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Full text of "Introduction to Quantum Mechanics with Applications to Chemistry"

Osmania Universitq uorur M 

; t-a^ I •«» Accession No. t$-?>T4-0 

ft No. ^ 60 » | ^ 

tle This book should be returned on or before the date- ■* 



Marked below 



INTRODUCTION 
QUANTUM MECHANICS 

With Applications to Chemistry 



LINUS PAUlteG, PhJ>>, Sc.D 

Professor of Chemistry, California institute of TecfonMotflT 
A#D 

E. BRIGHT WILSON,, Jr., Ph.D. 

Associate Professor of Chemistry, Harvard University 



INTERNATIONAL STUDENT EDITION 



McGRAW-HILL BOOK COMPAN^In^ ' ,U|H 

NEW YORK AND LONDON 

KOGAKUSHA COMPANY, LTD. 

TOKYO 



INTRODUCTION TO QUANTUM MECHANICS 

INTERNATIONAL STUDENT EDITION 

Exclusive rights by Kogakusha Co., Ltd. for manufacture and export 
from Japan. This book cannot be re-exported from the country to 
which a is consigned by Kogakusha Co., Ltd. or by McGraw-Hill Book 
Company, Inc. or any of Us subsidiaries. 

Copyright, 1935, by the 

McGraw-Hill Book Company, Inc. 

Jill rights reserved. This book, or 

parts :hei'Mf, iray net be reproduced 

in any form without permission of 

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PREFACE 

In writing this book we have attempted to produce a textbook 
of practical quantum mechanics for the chemist, the experi- 
mental physicist, and the beginning student of theoretical 
physics. The book is not intended to provide a critical discus- 
sion of quantum mechanics, nor even to presenita thorough 
survey of the subject. We hope that it does glvek lucid and 
easily understandable introduction to a limAdjporMon >» 
quantum-mechanical theorv; n*mv>K\ *ha( i J|twi ■ ■ U&*|# 
suggested by the rhiu«- ( .*..■/. / ., ; > n !■,-,., ffefrifetii^K of the 
alseussion oithe Schrodinger wave equal twi .• ft 15ie* problems 
which can be treated by means of it. The effort has been made 
to provide for the reader a means of equipping himself with a 
practical grasp of this subject, so that he can apply quantum 
mechanics to most of the chemical and physical problems which 
may confront him. 

The book is particularly designed for study by men without 
extensive previous experience with advanced mathematics, such 
as chemists interested in the subject because of its chemical 
applications. We have assumed on the part of the reader, in 
addition to elementary mathematics through the calculus, only 
some knowledge of complex quantities, ordinary differential 
equations, and the technique of partial differentiation. It 
may be desirable that a book written for the reader not adept 
at mathematics be richer in equations than one intended for 
the mathematician; for the mathematician can follow a sketchy 
derivation with ease, whereas if the less adept reader is to be 
led safely through the usually straightforward but sometimes 
rather complicated derivations of quantum mechanics a firm 
guiding hand must be kept on him. Quantum mechanics is 
essentially mathematical in character, and an understanding 
of the subject without a thorough knowledge of the mathematical 
methods involved and the results of their application cannot be 
obtained. The student not thoroughly trained in the theory 
of partial differential equations and orthogonal functions must 



iv PREFACE 

learn something of these subjects as he studies quantum mechan- 
ics. In order that he may do so, and that he may follow the 
discussions given without danger of being deflected from the 
course of the argument by inability to carry through some minor 
step, we have avoided the temptation to condense the various 
discussions into shorter and perhaps more elegant forms. 

After introductory chapters on classical mechanics and the 
old quantum theory, we have introduced the Schroding^r wave 
equation and its physical interpretation on a postulatory basis, 
and have then given in great detail the solution of the wave 
equation for important systems (harmonic oscillator, hydrogen 
atom) and the discussion of the wave functions and their proper- 
ties, omitting none of the mathematical steps except the most 
\}?JWJ*tesr?7 A, similarly detailed treatment has been given 
in the discussion o\ pert in 1 option 'ihreurj, the variation method, 
the structure of simple molecules, and, in general, iij. -•-, 
important section of the book. 

In order to limit the size of the book, we have omitted from 
discussion such advanced topics as transformation theory and 
general quantum mechanics (aside from brief mention in the 
last chapter), the Dirac theory of the electron, quantization 
of the electromagnetic field, etc. We have also omitted several 
subjects which are ordinarily considered as part of elementary 
quantum mechanics, but which are of minor importance to the 
chemist, such as the Zeeman effect and magnetic interactions in 
general, the dispersion of light and allied phenomena, and 
most of the theory of aperiodic processes. 

The authors are severally indebted to Professor A. Sommerfeld 
and Professors E. U. Condon and H. P. Robertson for their 
own introduction to quantum mechanics. The constant advice 
of Professor R. C. Tolman is gratefully acknowledged, as well 
as the aid of Professor P. M. Morse, Dr. L. E. Sutton, Dr. 
G. W. Wheland, Dr. L. 0. Brockway, Dr. J. Sherman, Dr. S. 
Weinbaum, Mrs. Emily Buckingham Wilson, and Mrs. Ava 
Helen Paulh^g. 

Linus Pauling. 
E. Bright Wilson, Jr. 

Pasadena, ^alif., 
Cambridge Mass., 
July, 1935. 



CONTENTS 

Page 

Preface , iii 

Chapter I 

SURVEY OF CLASSICAL MECHANICS 
Section 

1. Newton's Equations of Motion in the Lagrangian Form 2 

la. The Three-dimensional Isotropic Harmonic Oscillator. . 4 

16. Generalized Coordinates \. A . . . . 6 

lc. The Invariancc of the Equations of Motion in the Lagraib- 

gian Form 7 

Id. An Example: The Isotropic Harmonic Oscillator in Polar 

Coordinates 9 

lc. The Conservation of Angular Momentum . . . 11 

2. The Equations of Motion in the Hamiltonian Form 14 

2a. Generalized Momenta 14 

26. The Hamiltonian Function and Equations 16 

2c. The Hamiltonian Function and the Energy 16 

2d. A General Example 17 

3. The Emission and Absorption of Radiation . . 21 

4. Summary of Chapter 1 23 

Chapter II 
THE OLD QUANTUM THEORY 

5. The Origin of the Old Quantum Theory 25 

5a, The Postulates of Bohr 26 

56. The Wilson-Sommerfeld Rules of Quantization 28 

5c. Selection Rules. The Correspondence Principle 29 

6. The Quantization of Simple Systems 30 

6a. The Harmonic Oscillator. Degenerate States 30 

66. The Rigid Rotator 31 

6c. The Oscillating and Rotating Diatomic Molecule 32 

6d. The Particle in a Box 33 

6c. Diffraction by a Crystal Lattice 34 

7. The Hydrogen Atom 36 

7a. Solution of the Equations of Motion 36 

76. Application of the Quantum Rules. The Energy Levels . . 39 

7c. Description of the Orbits 43 

7a*. Spatial Quantization 45 

8. The Decline of the Old Quantum Theory 47 

v 



vi CONTENTS 

Section Page 

Chapter III 

THE SCHRODINGER WAVE EQUATION WITH THE 
HARMONIC OSCILLATOR AS AN EXAMPLE 

9. The Schrodinger Wave Equation 50 

9a. The Wave Equation Including the Time 53 

96. The Amplitude Equation 56 

9c. Wave Functions. Discrete and Continuous Sets of Charac- 
teristic Energy Values 58 

9d. The Complex Conjugate Wave Function ty*(x,t) . . . . 63 

10. The Physical Interpretation of the Wave Functions 63 

10a. ty*(x,t)ty(x,t) as a Probability Distribution Function. . 63 

106. Stationary States 64 

10c. Further Physical Interpretation. Average Values of 

Dynamical Quantities 65 

11. The Harmonic Oscillator in Wave Mechanics . . 67 

11a. Solution of the Wave Equation 67 

116. The Wave Functions for the Harmonic Oscillator and their 

Physical Interpretation ... 73 

lie. Mathematical Properties of the Harmonic Oscillator Wave 

Functions 77 

Chapter IV 

THE WAVE EQUATION FOR A SYSTEM OF POINT 

PARTICLES IN THREE DIMENSIONS 

12. The Wave Equation for a System of Point Particles 84 

12a. The Wave Equation Including the Time 85 

126. The Amplitude Equation 86 

12c. The Complex Conjugate Wave Function ty*(xi • ■ • z#, 88 

12d. The Physical Interpretation of the Wave Functions ... 88 

13 The Free Particle 90 

14. The Particle in a Box 95 

15. The Three-dimensional Harmonic Oscillator in Cartesian Coordi- 

nates 100 

16. Curvilinear Coordinates 103 

17. The Three-dimensional Harmonic Oscillator in Cylindrical Coordi- 

nates 105 

Chapter V 
THE HYDROGEN ATOM 

18. The Solution of the Wave Equation by the Polynomial Method 

and the Determination of the Energy Levels 113 

18a. The Separation of the Wave Equation. The Transla- 

tional Motion 113 

186. The Solution of the <p Equation 117 



CONTENTS Vit 

Suction Page 

18c. The Solution of the # Equation 118 

ISd. The Solution of the r Equation 121 

18c. The Energy Levels 124 

19. Legendre Functions and Surface Harmonics 125 

19a. The Legendre Functions or Legendre Polynomials 126 

196. The Associated Legendre Functions 127 

20. The Laguerre Polynomials and Associated Laguerre Functions . . 129 

20a. The Laguerre Polynomials 129 

206. The Associated Laguerre Polynomials and Functions ... 131 

21. The Wave Functions for the Hydrogen Atom 132 

21a. Hydrogen-like Wave Functions 132 

216. The Normal State of the Hydrogen Atom 139 

21c. Discussion of the Hydrogen-like Radial Wave Functions. . 142 
21d. Discussion of the Dependence of the Wave Functions 

on the Angles t? and <p 146 



Chapter VI 
PERTURBATION THEORY 

22. Expansions in Series of Orthogonal Functions 151 

23 ^First-order Perturbation Theory for a Non-degenerate Level . . . 156 
23a. A Simple Example: XhaJPexluikejj, gax mnnin Qs rillator. . 160 

9Zh An F, X arnp1p- Tin^JsTnrmfll H'^"™ Atom 162 

24. J£i£st-order Perturbation Theory for a Degenerate Level 165 

24a. An Example: Application of a Perturbation to a Hydrogen 

Atom 172 

25. Second-order Perturbation Theory 176 

25a. An Example: The Stark Effect of the Plane Rotator . . .177 



Chapter VII 

THE VARIATION METHOD AND OTHER APPROXIMATE 
METHODS 

26. The Variation Method 180 

26a. The Variational Integral and its Pr operties 180 

266. An Example: The Normal State of t he Helium A tom . . . 184 

26c. Application of the Variation Method to Other StaTes ... 186 

26d. Linear Variation Functions 186 

26c. A More General Variation Method 189 

27. Other Approximate Methods 191 

27a. A Generalized Perturbation Theory 191 

276. The Wentzel-Kramers-Brillouin Method 198 

27c. Numerical Integration 201 

27d. Approximation by the Use of Difference Equations .... 202 

27c. An Approximate Second-order Perturbation Treatment . . 204 



Via CONTENTS 

Suction Paqb 

Chapter VIII 

THE SPINNING ELECTRON AND THE PAULI EXCLUSION 
PRINCIPLE, WITH A DISCUSSION OF THE HELIUM ATOM 

28. The Spinning Electron 207 

29. The Helium Atom. The Pauli Exclusion Principle 210 

29a. The Configurations ls2s and ls2p 210 

296. The Consideration of Electron Spin. The Pauli Exclusion • 

Principle 214 

29c. The Accurate Treatment of the Normal Helium Atom. . . 221 

29d. Excited States of the Helium Atom 225 

29e. The Polarizability of the Normal Helium Atom 226 

Chapter IX 
MANY-ELECTRON ATOMS 

30. Slater's Treatment of Complex Atoms 230 

30a. Exchange Degeneracy 230 

306. Spatial Degeneracy 233 

30c. Factorization and Solution of the Secular Equation. . . 235 

30d. Evaluation of Integrals ... * 239 

30e. Empirical Evaluation of Integrals. Applications. . . . 244 

31. Variation Treatments for Simple Atoms 240 

31a. The Lithium Atom and Three-electron Ions . .... 247 
316. Variation Treatments of Other Atoms. . . 249 

32. The Method of the Self-consistent Field 250 

32a. Principle of the Method 250 

326. Relation of the Self-consistent Field Method to the Varia- 
tion Principle ^ . . . 252 

32c. Results of the Self-consistent Field Method 254 

33. Other Methods for Many-electron Atoms 256 

33a. Semi-empirical Sets of Screening Constants 256 

336. The Thomas-Fermi Statistical Atom 257 

Chapter X 
THE ROTATION AND VIBRATION OF MOLECULES 

34. The Separation of Electronic and Nuclear Motion 259 

35. The Rotation and Vibration of Diatomic Molecules 263 

35a. The Separation of Variables and Solution of the Angular 

Equations 264 

356. The Nature of the Electronic Energy Function 266 

35c. A Simple Potential Function for Diatomic Molecules . . . 267 
35d. A More Accurate Treatment. The Morse Function . . . 271 

36. The Rotation of Polyatomic Molecules 275 

36a. The Rotation of Symmetrical-top Molecules 275 

366. The Rotation of Unsymmetrical-top Molecules 280 



CONTENTS ix 

Section Page 

37. The Vibiation of Polyatomic Molecules 282 

37a. Normal Coordinates in Classical Mechanics 282 

376. Normal Coordinates in Quantum Mechanics 288 

38. The Rotation of Molecules in Crystals 290 

Chapter XI 

PERTURBATION THEORY INVOLVING THE TIME, THE 

EMISSION AND ABSORPTION OF RADIATION, AND THE 

RESONANCE PHENOMENON 

39. The Treatment of a Time-dependent Perturbation by the Methoa 

of Variation of Constants .... 294 

39a. A Simple Example .... 296 

40. The Emission and Absorption of Radiation. V .... 299 

40a. The Einstein Transition Probabilities .... 299 

406. The Calculation of the Einstein Transition Probabilities 

by Perturbation Theory 302 

40c. Selection Rules and Intensities for the Harmonic Oscillator 306 
40d. Selection Rules and Intensities for Surface -harmonic Wave 

Functions 306 

40e. Selection Rules and Intensities for the Diatomic Moleculev. 

The Franck-Condon Principle 309 

40/. Selection Rules and Intensities for the Hydrogen Atom . .312 
40a. Even and Odd Electronic States and their Selection Rules. 313 

41. The Resonance Phenomenon 314 

41a. Resonance in Classical Mechanics 315 

416. Resonance in Quantum Mechanics 318 

41c. A Further Discussion of Resonance 322 

Chapter XII 
THE STRUCTURE OF SIMPLE MOLECULES 

42. The Hydrogen Molecule-ion 327 

42a. A Very Simple Discussion 327 

426. Other Simple Variation Treatments 331 

42c. The Separation and Solution of the Wave Equation .... 333 

42a\ Excited States of the Hydrogen Molecule-ion 340 

43. The Hydrogen Molecule 340 

43a. The Treatment of Heitler and London. 340 

436. Other Simple Variation Treatments . . . 345 

43c. The Treatment of James and Coolidge 349 

43d. Comparison with Experiment 351 

43e. Excited States of the Hydrogen Molecule 353 

43/. Oscillation and Rotation of the Molecule. Ortho and 

Para Hydrogen 355 

44. The Helium Molecule-ion He£ and the Interaction of Two Normal 

Helium Atoms 358 

44a. The Helium Molecule-ion Hef 358 



INTRODUCTION TO QUANTUM 
MECHANICS 

CHAPTER I 
SURVEY OF CLASSICAL MECHANICS 

The subject of quantum mechanics constitutes the most recent 
step in the very old search for the general laws : jroyjyming the 
motion of matter. For a long time investigators confined their 
efforts to studying the dynamics of bodies of macroscopic dimen- 
sions, and while the science of mechanics remained in that 
stage it was properly considered a branch of physics. Since 
the development of atomic theory there has been a change of 
emphasis. It was recognized that the older laws are not correct 
when applied to atoms and electrons, without considerable 
modification. Moreover, the success which has been obtained 
in making the necessary modifications of the older laws has also 
had the result of depriving physics of sole claim upon them, since 
it is now realized that the combining power of atoms and, in 
fact, all the chemical properties of atoms and molecules are 
explicable in terms of the laws governing the motions of the 
electrons and nuclei composing them. 

Although it is the modern theory of quantum mechanics in 
which we are primarily interested because of its applicatiqnsjto 
chemical problems, it is desirable for us first to discuss briefly 
the background of classical mechanics from which it was devel- 
oped. By so doing we not only follow to a certain extent the 
historical development, but we also introduce in a more familiar 
form many concepts which are retained in the later theory. We 
shall also treat certain problems in the first few chapters by the 
methods of the older theories in preparation for their later treat- 
ment by quantum mechanics. It is for this reason that the 
student is advised to consider the exercises of the first few 
chapters carefully and to retain for later reference the results 
which are secured. 

1 



2 SURVEY OF CLASSICAL MECHANICS [1-1 

In the first chapter no attempt will be made to give any parts 
of classical dynamics but those which are useful in the treatment 
of atomic and molecular problems. With this restriction, we 
have felt justified in omitting discussion of the dynamics of rigid 
bodies, non-conservative systems, non-holonomic systems, sys- 
tems involving impact, etc. Moreover, no use is made of 
Hamilton's principle or of the Hamilton-Jacobi partial differential 
equation. By thus limiting the subjects to be discussed, it is 
possible to give in a short chapter a thorough treatment of 
Newtonian systems of point particles. 

1. NEWTON'S EQUATIONS OF MOTION IN THE LAGRANGIAN 

FORM 

The earliest formulation of dynamical laws of wide application 
is that of Sir Isaac Newton. If we adopt the notation Xi, y t , Zi 
for the three Cartesian coordinates of the ith particle with 
mass Wi, Newton's equations for n point particles are 

m x Xi = X t) ) 

m x iji = Y if } i = 1, 2, • • • , n, (1-1) 



m x yi = i if / 
miii = Z X) ) 



where X», Y x , Z t are the three components of the force acting on 
the ^th particle. There is a set of such equations for each 
particle. Dots refer to differentiation with respect to time, so 
that 

d 2 r- 

*< - w- ™ 

By introducing certain familiar definitions we change Equation 
1-1 into a form which will be more useful later. We define as 
the kinetic energy T (for Cartesian coordinates) the quantity 

T = Mm x {±\ + y\ + i\) + • • • + V 2 m n {xl + yl + zl) 

n 

= h X mii± < + y? + #• ( 1_3 ) 

If we limit ourselves to a certain class of systems, called conserva- 
tive systems, it is possible to define another quantity, the potential 
energy V, which is a function of the coordinates Xii/iZi • • • 
x n y n z n of all the particles, such that the force components acting 



1-1] NEWTON'S EQUATIONS OF MOTION 3 

on each particle are equal to partial derivatives of the potential 
energy with respect to the coordinates of the particle (with 
negative sign) ; that is, 

A{ = — - — > 

Y< = —,) i = 1, 2, • • • , n. (1-4) 

It is possible to find a function V which will express in this manner 
forces of the types usually designated as mechanical, electrostatic, 
and gravitational. Since other types of forces (such as electro- 
magnetic) for which such a potential-energy function cannot 
be set up are not important in chemical applications, we shall 
not consider them in detail. 

With these definitions, Newton's equations become 

(l-5a) 

(1-56) 

(l-5c) 

There are three such equations for every particle, as before. 
These results are definitely restricted to Cartesian coordinates; 
but by introducing a new function, the Lagrangian function L f 
defined for Newtonian systems as the difference of the kinetic 
and potential energy, 

L = L(xi, y h Zi, • • • , x n , y n , z n , x h • • • , z n ) = 

T - V, (1-6) 

we can throw the equations of motion into a form which we shall 
later prove to be valid in any system of coordinates (Sec. lc). 
In Cartesian coordinates T is a function of the velocities 
xi 9 • • • , i n only, and for the systems to which our treatment 
is restricted V is a function of the coordinates only; hence the 
equations of motion given in Equation 1-5 on introduction of 
the function L assume the form 



d^dT dV _ 
dt d±i dZi 


= 0, 


d dT dV _ 
dt dy { dy { 
d dT dV 
dt dii dZi 


= 0, 
= 0. 



SURVEY OF CLASSICAL MECHANICS 


d dL _ 
dt dxi 


dL 

dXi 


= o,\ 




d dL _ 
eft d^t 
d dL __ 
dt dii 


dL 
dyi 
dL 

dZi 


= 0,1 


i = 1, 2, • • • , 7 



[I-la 



(1-7) 



In the following paragraphs a simple dynamical system is 
discussed by the use of these equations. 

la. The Three-dimensional Isotropic Harmonic Oscillator. — 
As an illustration of the use of the equations of motion in this 
form, we choose a system which has played a very important 
part in the development of quantum theory. This is the 
farngnij^osctJjMor, a particle bound to an equilibrium position by 
a force which increases in magnitude linearly with its distance 
r from the point. In the three-dimensional isotropic harmonic 
oscillator this corresponds to a potential function }ikr 2 , represent- 
ing a force of magnitude kr acting in a negative direction; i.e., 
from the position of the particle to the origin, k is called the 
force constant or Hooke's-law constant. Using Cartesian coordi- 
nates we have 

L = y 2 m(x> + y 2 + * 2 ) - V2k(x> + y 2 + z 2 ), (1-8) 
whence 

-rXmx) + kx = mx + kx = 0,] 

my + ky =0,/ (1~9) 

m'z + kz = O.j 

Multiplication of the first member of Equation 1-9 by x gives 

mi tt - - kx % (1_10) 

or 

2 m dt 2 k dt ' U ll) 

which integrates directly to 

%mx 2 = -Y 2 kx 2 + constant. (1-12) 

The constant of integration is conveniently expressed as x /ikx\. 



I-la] NEWTON'S EQUATIONS OF MOTION 

Hence 



dx ___ fie 
dl " \m 



or 

and similarly 



(*l - * 2 ), (1-13) 

or, on introducing the expression \-K 2 mv\ in place of the force 
constant k y 

which on integration becomes 

2wvot + 5 X = sin -1 — 

x = x sin (2irM + $*), (1-14) 

2/ = t/o sin (aryrf + & y )\ (1-15) 

2 = Zq sin (2tt^ < + 5.)- / 
In these expressions x , y , z , 8 X , 8 y , and 8 Z are constants of 
integration, the values of which determine the motion in any 
given case. The quantity v is related to the constant of the 
restoring force by the equation 

4r*mv$ = Jfc, (1-16) 

so that the potential energy may be written as 

V = 2^mv'ir\ (1-17) 

As shown by the equations for x, y, and z, p is the frequency of 
the motion. It is seen that the particle may be described as 
carrying out independent harmonic oscillations along the x, y } 
and z axes, with different amplitudes Xo, t/o, and Zq and different 
phase angles 8 X , b yy and 8 Z , respectively. 

The energy of the system is the sum of the kinetic energy and 
the potential energy, and is thus equal to 

}im(x 2 + y 2 + z 2 ) + 2jr 2 mvl(x 2 + y 2 + z 2 ). 

On evaluation, it is found to be independent of the time, with the 
value 2Tr 2 mvl(xl + y\ + zj) determined by the amplitudes of 
oscillation. 

The one-dimensional harmonic oscillator, restricted to -motion 
along the x axis in accordance with the potential function 
V = }ikx 2 = 2w 2 mv%x 2 , is seen to carry out harmonic oscillations 



6 SURVEY OF CLASSICAL MECHANICS [I-lb 

along this axis as described by Equation 1-14. Its total energy- 
is given by the expression 2w 2 mvlxl 

lb. Generalized Coordinates. — Instead of Cartesian coordi- 
nates xi, 2/1, 2i, • • • , x n , y n , z n , it is frequently more convenient 
to use some other «et of coordinates to specify the configuration 
of the system. For example, the isotropic spatial harmonic 
oscillator already discussed might equally well be described using 
polar coordinates; again, the treatment of a system composed of 
two attracting particles in space, which will be considered 
later, would be very cumbersome if it were necessary to use 
rectangular coordinates. 

If we choose, any set of 3n coordinates, which we shall always 
assume to be independent and at the same time sufficient in 
number to specify completely the positions of the particles of 
the system, then there will in general exist 3n equations, called 
the equations of transformation, relating the new coordinates 
qn to the set of Cartesian coordinates x t , y tf z i} 



(1-18) 



There is such a set of three equations for each particle i. The 
functions / t , g i} hi may be functions of any or all of the Zn new 
coordinates q k , so that these new variables do not necessarily 
split into sets which belong to particular particles. For example, 
in the case of two particles the six new coordinates may be the 
three Cartesian coordinates of the center of mass together 
with the polar coordinates of one particle referred to the other 
particle as origin. 

As is known from the theory of partial differentiation, it is 
possible to transform derivatives from one set of independent 
variables to another, an example of this process being 

dt dq t dt "*" dq 2 dt "*" "*" dq 3n IF' ^~ iya ^ 

This same equation can be put in the much more compact form 

3n 

**«* (1-196) 



*i = /i(<?l> 02, ' * 


• > 03«), 


Vi = Vifal, 02, ' ' 


• , ?3n), 


Zi = hi(q ly q 2 , ' ' 


• , Q*n). 



XT' dZi . 



I-lc] NEWTON'S EQUATIONS OF MOTION 7 

This gives the relation between any Cartesian component of 
velocity and the time derivatives of the new coordinates. Similar 
relations, of course, hold for t/» and Zi for any particle. The 
quantities % by analogy with £», are called generalized velocities, 
even though they do not necessarily have the dimensions of 
length divided by time (for example, qj may be an angle). 

Since partial derivatives transform in just the same manner, 
we have 

_£Z" = _^Z^ _ ^Z d l} _ . . . _ ^z^? 

dqj dxi dqj dyi dqj dz n dq s 

= " 2(a*, Jg, + tyi "4/ + a£ wi) = Qi ' (1_20) 

Since Q 3 is given by an expression in terms of V and qj which is 
analogous to that for the force Xi in terms of V and x i} it is called 
a generalized force. 

In exactly similar fashion, we have 

*! = ^2(*! d Ai + wdj A wdzA ( . 

dq 3 - 2Z\d±i dqj "*" dy { dqj "*" di< dqj)' } 

lc. The Invariance of the Equations of Motion in the Lagran- 
gian Form. — We are now in a position to show that when New- 
ton's equations are written in the form given by Equation 1-7 
they are valid for any choice of coordinate system. For this 
proof we shall apply a transformation of coordinates to Equa- 
tions 1-5, using the methods of the previous section. Multiplica- 

dx ' du ' 

tion of Equation l-5a by — *> of 1-56 by -~) etc., gives 

dqj dqj 

dxiddT^ ^Z^i ^ ^ 

dqj dt d±i dxi dqj ' 

dzidW dVdXi = 0( 

dqj dt d± 2 dx 2 dqj '} (1-22) 



dXnddT^ dVdXn = Q | 
dqj dt dx n dx n dq, * * 



with similar equations in y and z. Adding all of these together 
gives 



8 SURVEY OF CLASSICAL MECHANICS [I-lc 

2l \dqj dt d±i "*" dqi dt dj/i ^ dq, dt diij "*" dq, ' K } 

where the result of Equation 1-20 has been used. In order to 
reduce the first sum, we note the following identity, obtained by- 
differentiating a product, 

dXid/dT\ = d/dTdxA _ dTd/dxA 
dqjdtydXi) dt\dXidqi) dXidt\dqj)' ( ) 

From Equation 1-1% we obtain directly 

ft = 1/ «-**> 

Furthermore, because the order of differentiation is immaterial, 
we see that 

d/dXi\ = ^±_/<tei\ d = N^A/^A- 
cttUfc/ <Zl dq k \Sqi ) Qk 2j dq\dq k ) qk 

& = 1 *-l 

By introducing Equations 1-26 and 1-25 in 1-24 and using the 
result in Equation 1-23, we get 



dTdyi 
dy { dqj 



2(d/dT d±i , dTdyi dT diA __ /dT dxt 
\dt\dXi dqj + dyi dqj + di; dqj) \dxi dqj 

which, in view of the results of the last section, reduces to 
ddT dT dV_ 

Finally, the introduction of the Lagrangian function L = T — V, 
with V a function of the coordinates only, gives the more compact 
form 



I-ld] NEWTON'S EQUATIONS OF MOTION 9 

(It is important to note that L must be expressed as a function 
of the coordinates and their first time-derivatives.) 

Since the above derivation could be carried out for any value 
of j, there are 3n such equations, one for each coordinate q,-. 
They are called the equations of motion in the Lagrangian form 
and are of great importance. The method by which they were 
derived shows that they are independent of the coordinate 
system. 

We have so far rather limited the types of systems considered, 
but Lagrange's equations are much more general than we have 
indicated and by a proper choice of the function L nearly all dynami- 
cal problems can be treated with their use. These equations are 
therefore frequently chosen as the fundamental postulates of 
classical mechanics instead of Newton's laws. 

Id. An Example : The Isotropic Harmonic Oscillator in Polar 
Coordinates. — The example which we have treated in Section la 
can equally well be solved by the use of polar coordinates r, 
#, and <p (Fig. 1-1) . The equations of transformation correspond- 
ing to Equation 1-18 are 

x = rsin#cos^,) 

y = r sin # sin tp, \ (1-30) 

z = rcostf. j 

With the use of these we find for the kinetic and potential energies 
of the isotropic harmonic oscillator the following expressions: 

T = ~ m(x 2 + y 2 + z 2 ) = | (r 2 + rW + r 2 sin 2 * <p 2 )) (1 _ 31) 
V = 27r 2 mv 2 r 2 , j 

and 
L = T - V = ~(r 2 + r 2 & 2 + r 2 sin 2 #<p 2 ) - 2nr 2 mv\r 2 . (1-32) 

The equations of motion are 

£ $k _ M = ^(mr 2 ^) - mr 2 sin a cos &<p 2 = 0, (1-34) 

at $# oft at 

t*-z-. :r = T-(wr) — mr# 2 — mr sin 2 &v> 2 + Ar 2 mvlr «* 0. 

dtdr dr dC ' ° 

(1-35) 



10 



SURVEY OF CLASSICAL MECHANICS 



I-id 



In Appendix II it is shown that the motion takes place in a 
plane containing the origin. This conclusion enables us to 
simplify the problem by making a change of variables. Let us 
introduce new polar coordinates r, #', x such that at the time 
t = the plane determined by the vectors r and v, the position 
and velocity vectors of the particle at t = 0, is normal to the new 
z' axis. This transformation is known in terms of the old set of 
coordinates if two parameters # and v? , determining the position 
of the axis z' in terms of the old coordinates, are given (Fig. 1-2). 




Fig. 1-1. — The relation of polar coor- 
dinates r, t? , and tp to Cartesian axes. 



Fig. 1-2. — The rotation of axes. 



In terms of the new coordinates, the Lagrangian function L 
and the equations of motion have the same form as previously, 
because the first choice of axis direction was quite arbitrary. 
However, since the coordinates have been chosen so that the 
plane of the motion is the x'y' plane, the angle &' is always equal 
to a constant, v/2. Inserting this value of & in Equation 1-33 
and writing it in terms of x instead of <p, we obtain 



dt 



(mr 2 x) = 0, 



which has the solution 



mr 2 \ = p X) a constant. 
The r equation, Equation 1-35, becomes 

-n(™>r) — mry} + 4r 2 mvlr = 0, 



(1-36) 



(1-37) 



Me] NEWTON'S EQUATIONS OF MOTION 11 

or, using Equation 1-37, 

J t (mr) ~£i + 4r*m,tfr = 0, (1-38) 

an equation differing from the related one-dimensional Cartesian- 
coordinate equation by the additional term — p$/mr z which 
represents the centrifugal force. 

Multiplication by f and integration with respect to the time 
gives 

f2 = -^ ~ 47r2 "° r2 + b > (1 " 39) 



so that r = ( — ^- 9 - \>w 2 v 2 r 2 + b) • 
\ m 2 r 2 u / 

This can be again integrated, to give 



t - U 



P rdr 



2J (a 



dx 

+ bx + cx 2 Y 



in which x = r 2 , a = — p|/m 2 , b is the constant of integration in 
Equation 1-39, and c = —iw 2 vl. This is a standard integral 
which yields the equation 



-g— 2 {& + A sin 4irv (t - to)}, 



with A given by 



A 



= ^ 2 " 



16?r 2 vgp 2 



m L 



We have thus obtained the dependence of r on the time, and 
by integrating Equation 1-37 we could obtain xasa function of 
the time, completing the solution. Elimination of the time 
between these two results would give the equation of the orbit, 
which is an ellipse with center at the origin. It is seen that the 
constant v again occurs as the frequency of the motion. 

le. The Conservation of Angular Momentum. — The example 
worked out in the previous section illustrates an important 
principle of wide applicability, the principle of the conservation 
of angular momentum. 



12 



SURVEY OF CLASSICAL MECHANICS 



U-le 



Equation 1-37 shows that when x is the angular velocity of the 
particle about a fixed axis z' and r is the distance of the particle 
from the axis, the quantity p x = mr 2 \ is a constant of the motion. * 
This quantity is called the angular momentum of the particle 
about the axis z'. 

It is not necessary to choose an axis normal to the plane of the 
motion, as z' in this example, in order to apply the theorem. 
Thus Equation 1-33, written for arbitrary direction z, is at once 
integrable to 

rar 2 sin 2 &<p = p^ a constant. (1-40) 

Here r sin & is the distance of the particle from the axis z, so that 
the left side of this equation is the angular momentum about the 
axis z. 2 It is seen to be equal to a constant, p*. 




Fig. 1-3. — Figure showing the relation between dx, d&, and d<p. 

In order to apply the principle, it is essential that the axis of 
reference be a fixed axis. Thus the angle & of polar coordinates 
has associated with it an angular momentum p# = mrH about 
an axis in the xy plane, but the principle of conservation of 
angular momentum cannot be applied directly to this quantity 
because the axis is not, in general, fixed but varies with ^. A 
simple relation involving p* connects the angular momenta 

1 The phrase a constant of the motion is often used in referring to a constant 
of integration of the equations of motion for a dynamical system. 

2 This is sometimes referred to as the component of angular momentum 
along the axis z. 



I-le] NEWTON'S EQUATIONS OF MOTION 13 

p x and p<p about different fixed axes, one of which, p x , relates 
to the axis normal to the plane of the motion. This is 

p x dx = P*d& + pyd<p, (1-41) 

an equation easily derived by considering Figure 1-3. The 
sides of the small triangle have the lengths r sin dd(p, rdx, and 
rd&. Since they form a right triangle, these distances are 
connected by the relation 

r\d%Y = r 2 sin 2 #(d<p) 2 + r 2 (d#) 2 , 

which gives, on introduction of the angular velocities x, <P, and # 
and multiplication by m/dt, 

mr 2 xdx = mr 2 sin 2 &<pd<p + mr 2 $d&. 

Equation 1-41 follows from this and the definitions of p X} p*, 
and p v . 

Conservation of angular momentum may be applied to more 
general systems than the one described here. It is at once 
evident that we have not used the special form of the potential- 
energy expression except for the fact that it is independent of 
direction, since this function enters into the r equation only. 
Therefore the above results are true for a particle moving in 
any spherically symmetric potential field. 

Furthermore, we can extend the theorem to a collection of 
point particles interacting with each other in any desired way 
but influenced by external forces only through a spherically 
symmetric potential function. If we describe such a system by 
using the polar coordinates of each particle, the Lagrangian 
function is 

n 

L = V 2 X m ^ + r & + r $ sin2 *&f) - V. (1-42) 
i = i 

Instead of <pi, <p 2 , • • • , <p n) we now introduce new angular 
coordinates a, 0, • • • , k given by the linear equations 

<Pi = a + bip + - • • + kiK 9 \ 

<p 2 = a + b 2 p + ' ' ' + k 2 K,l ,. . ox 

> (1-43) 
>\ 

<Pn = a +bnP + * ' * +k n K.) 

The vaiues given the constants b u • • • , k n are unimportant so 
long as they make the above set of equations mutually independ- 



14 SURVEY OF CLASSICAL MECHANICS [I-2a 

ent. a is an angle about the axis z such that if a is increased 
by Aa, holding /?, • * • , k constant, the effect is to increase each 
ipi by Aa, or, in other words, to rotate the whole system of particles 
about z without changing their mutual positions. By hypothesis 
the value of V is not changed by such a rotation, so that V is 
independent of a. We therefore obtain the equation 

daL_6L = d_3T = 

dt da da dt da U **' 

Moreover, from Equation 1-42 we derive the relation 

2Sr£ -2-'* -•*■»- < i -«> 



dT 
da 



t = l 



Hence, calling the distance r t sin d x of the tth particle from the 
z axis pi, we obtain the equation 

n 

^m i p^<p l = constant. (1-46) 

i = i 

This is the more general expression of the principle of the con- 
servation of angular momentum which we were seeking. In 
such a system of many particles with mutual interactions, as, 
for example, an atom consisting of a number of electrons and a 
nucleus, the individual particles do not in general conserve 
angular momentum but the aggregate does. 

The potential-energy function V need be only cylindrically 
symmetric about the axis z for the above proof to apply, 
since the essential feature was the independence of V on the angle 
a about z. However, in that case z is restricted to a particular 
direction in space, whereas if V is spherically symmetric the 
theorem holds for any choice of axis. 

Angular momenta transform like vectors, the directions of the 
vectors being the directions of the axes about which the angular 
momenta are determined. It is customary to take the sense 
of the vectors such as to correspond to the right-hand screw rule. 

3. THE EQUATIONS OF MOTION IN THE HAMILTONIAN FORM 

2a. Generalized Momenta. — In Cartesian coordinates the 
momentum related to the direction x k is m k x kj which, since V is 



I-2a] EQUATIONS OF MOTION IN HAMILTONIAN FORM 15 

restricted to be a function of the coordinates only, can be written 
as 

^ = 1 = 1' * = l,2,---,3n. (2-1) 

Angular momenta can likewise be expressed in this manner. 
Thus, for one particle in a spherically symmetric potential field, 
the angular momentum about the z axis was defined in Section le 
by the expression 

ZV = mp 2 <p = mr 2 sin 2 &<p. (2-2) 

Reference to Equation 1-31, which gives the expression for the 
kinetic energy in polar coordinates, shows that 

* - * - w (2 ~ 3) 

Likewise, in the case of a number of particles, the angular 
momentum conjugate to the coordinate a is 

as shown by the discussion of Equation 1-46. By extending 
this to other coordinate systems, the generalized momentum pk 
conjugate to the coordinate q k is defined as 

Vk = Jg, k = 1, 2, • • • , 3n. (2-5) 

The form taken by Lagrange's equations (Eq. 1-29) when the 
definition of p* is introduced is 

p k = ^, k = 1, 2, • • • , 3n, (2-6) 

so that Equations 2-5 and 2-6 form a set of 6n first-order dif- 
ferential equations equivalent to the 3n second-order equations 
of Equation 1-29. 

— being in general a function of both the q'a and q's, the 
oqk 

definition of p k given by Equation 2-5 provides 3n relations 

between the variables q k) <ik, and pk, permitting the elimination 

of the 3n velocities q k , so that the system can now be described 

in terms of the 3n coordinates qk and the 3n conjugate momenta 



16 SURVEY OF CLASSICAL MECHANICS (I-2c 

p k . Hamilton in 1834 showed that the equations of motion can 
in this way be thrown into an especially simple form, involving 
a function H of the p k 's and q k s called the Hamiltonian function. 
2b. The Hamiltonian Function and Equations. — For con- 
servative systems. 1 we shall show that the function H is the total 
energy (kinetic plus potential) of the system, expressed in terms 
of the pk's and q k $. In order to have a definition which holds 
for more general systems, we introduce H by the relation 

3n 

H = ]£?*& - L(q k , q k ). (2-7) 

Although this definition involves the velocities q k) H may be made 
a function of the coordinates and momenta only, by eliminating 
the velocities through the use of Equation 2-5. From the 
definition we obtain for the total differential of H the equation 

3n 3n 3n 3n 

dH = ^Pkdq k + 2«*dp* - 2jdq~ k dqk " 2^^*' ^ 2 " 8 ^ 

or, using the expressions for p k and p k given in Equations 2-5 and 
2-6 (equivalent to Lagrange's equations), 

3n 

dH = 2)(g*dpib - Pkdqk), (2-9) 

fc = l 

whence, if // is regarded as a function of the q k s and p k s, we 
obtain the equations 



dH _ 
dpk 


4k. 


> 


dH _ 
dq k 


— 


Pk, 



1, 2, • • • , 3n. (2-10) 



These are the equations of motion in the Hamiltonian or canonical 
form. ~ K 

2c. The Hamiltonian Function and the 'Energy. — Let us con- 
sider the time dependence of H for a conservative system. We 
have 

1 A conservative system is a system for which H does not depend explicitly 
on the time t. We have restricted our discussion to conservative systems by 
assuming that the potential function V does not depend on t. 



1-81 THE EMISSION AND ABSORPTION OF RADIATION 21 

coordinate but only its derivative. Such a coordinate is cahea a cyclic 
coordinate. 

3. THE EMISSION AND ABSORPTION OF RADIATION 

The classical laws of mechanical and electromagnetic theory 
permit the complete discussion of the emission and absorption of 
electromagnetic radiation by a system of electrically charged 
particles. In the following paragraphs we shall outline the 
results of this discussion. It is found that these results are not 
in agreement with experiments involving atoms and molecules; 
it was, indeed, just this disagreement which was the principal 
factor in leading to the development of the Bohr theory of the 
atom and later of the quantum mechanics. Even at the present 
time, when an apparently satisfactory theoretical treatment of 
dynamical systems composed of electrons and nuclei is provided 
by the quantum mechanics, the problem of the. emission and 
absorption of radiation still lacks a satisfactory solution, despite 
the concentration of attention on it by the most able theoretical 
physicists. It will be shown in a subsequent -chapter howeyer, 
that, despite our lack of a satisfactory conception of the nature 
of electromagnetic radiation, equations similar to the classical 
equations of this section can be formulated which represent 
correctly the emission and absorption of radiation by' atomic 
systems to within the limits of error of experiment. 

According to the classical theory the rate of emission of radiant 
energy by an accelerated particle of electric charge e is 

dt 3c 3 ' 

dE 
in which — -^- is the rate at which the energy E of the particle 

is converted into radiant energy, v is the acceleration of the 
particle, and c the velocity of light. 

Let us first consider a system of a special type, in which a 
particle of charge e carries out simple harmonic oscillation 
with frequency v along the x axis, according to the equation 

x = x cos 2rvt. (3-2) 

Differentiating this expression, assuming that z* is independent 
of the time, we obtain for the acceleration the value 

v ss x == — 4irV£o cos %cvt. 



22 SURVEY OF CLASSICAL MECHANICS [I-» 

The average rate of emission of radiant energy by such a system 
is consequently 

~~dt = 3c* ' (3 ^ } 

inasmuch as the average value cos 2 2rvt over a cycle is "one-half. 
As a result of the emission of energy, the amplitude x of the 
motion will decrease with time; if the fractional change in 
energy during a cycle of the motion is small, however, this equa- 
tion retains its validity. 

The radiation emitted by such a system has the frequency v of the 
emitting system. It is plane-polarized, the plane of the electric 
vector being the plane which includes the x axis and the direction 
of propagation of the light. 

In case that the particle carries out harmonic oscillations along 
all three axes x, y, and z, with frequencies v x> v Vt and v z and 
amplitudes (at a given time) Xo, yo, and z 0f respectively, the total 
rate of emission of radiant energy will be given as the sum of 
three terms similar to the -right side of Equation 3-4, one giving 
the rate of emission of energy as light of frequency v Xl one of 
v V) and one of v t . 

If the motion of the particle is not simple harmonic, it can be 
represented by a Fourier series or Fourier integral as a sum or 
integral of harmonic terms similar to that of Equation 3-2; 
light of frequency characteristic of each of these terms will then 
be emitted at a rate given by Equation 3-4, the coefficient of the 
Fourier term being introduced in place of x . 

The emission of light by a system composed of several inter- 
acting electrically charged particles is conveniently discussed in 
the following way. A Fourier analysis is first made of the 
motion of the system in a given state to resolve it into harmonic 
terms. For a given term, corresponding to a given frequency 
of motion v y the coefficient resulting from the analysis (which is a 
function of the coordinates of the particles) is expanded as a 
power series in the quantities Xi/\ • • • , z n /\ in which xi, 
• • • , z n are the coordinates of the particles relative to some 
origin (such as the center of mass) and X = c/v is the wave length 
of the radiation with frequency v. The term of zero degree in 
this expansion is zero, inasmuch as the electric charge of the 
system does not change with time. The term of first degree 
involves, in addition to the harmonic function of the time, only 



1-4] SUMMARY OF CHAPTER I 23 

a function of the coordinates. The aggregate of these first- 
degree terms in the coordinates with their associated time factors, 
summed over all frequency values occurring in the original 
Fourier analysis, represents a dynamical quantity known as the 
electric moment of the system, a vector quantity P defined as 

P = 2)*r< f (3-5) 

i 

in which r» denotes the vector from the origin to the position of 
the ith. particle, with charge e». Consequently to this degree of 
approximation the radiation emitted by a system of several 
particles can be discussed by making a Fourier analysis of the 
electric moment P. Corresponding to each term of frequency v 
in this representation of P, there will be emitted radiation of 
frequency v at a rate given by an equation similar to Equation 
3-4, with exo replaced by the Fourier coefficient in the electric- 
moment expansion. The emission of radiation by this mechanism 
is usually called dipole emission, the radiation itself sometimes 
being described as dipole radiation. 

The quadratic terms in the expansions in powers of Xi/\ 
• • • , Zn/X form a quantity Q called the quadrupole mo'ment 
of the system, and higher powers form higher moments. The rate 
of emission of radiant energy as a result of the change of quadru- 
pole and higher moments of an atom or molecule is usually 
negligibly small in comparison with the rate of dipole emission, 
and in consequence dipole radiation alone is ordinarily discussed. 
Under some circumstances, however, as when the intensity of 
dipole radiation is zero and the presence of very weak radiation 
can be detected, the process of quadrupole emission is important. 

4. SUMMARY OF CHAPTER I 

The purpose of this survey of classical mechanics is twofold: 
first, to indicate the path whereby the more general formulations 
of classical dynamics, such as the equations of motion of Lagrange 
and of Hamilton, have been developed from the original equations 
of Newton; and second, to illustrate the application of these 
methods to problems which are later discussed by quantum- 
mechanical methods. 

In carrying out the first purpose, we have discussed Newton's 
equations in Cartesian coordinates and then altered their form by 



24 SURVEY OF CLASSICAL MECHANICS [M 

the introduction of the kinetic and potential energies. By 
defining the Lagrangian function for the special case of Newtonian 
systems and introducing it into the equations of motion, Newton's 
equations were then thrown into the Lagrangian form. Follow- 
ing an introductory discussion of generalized coordinates, the 
proof of the general validity of the equations of motion in the 
Lagrangian form for any system of coordinates has been given; 
and it has also been pointed out that the Lagrangian form 
of the equations of motion, although we have derived it from the 
equations of Newton, is really more widely applicable than 
Newton's postulates, because by making a suitable choice of the 
Lagrangian function a very wide range of problems can be 
treated in this way. 

In the second section there has been derived a third form for 
the equations of motion, the Hamiltonian form, following the 
introduction of the concept of generalized momenta, and the rela- 
tion between the Hamiltonian function and the energy has been 
discussed. 

In Section 3 a very brief discussion of the classical theory of 
the radiation of energy from accelerated charged particles has 
been given, in order to have a foundation for later discussions 
of this topic. Mention is made of both dipole and quadrupole 
radiation. 

Finally, several examples (which are later solved by the use of 
quantum mechanics), including the three-dimensional harmonic 
oscillator in Cartesian and in polar coordinates, have been 
treated by the methods discussed in this chapter. 

General References on Classical Mechanics 

W. D. MacMillan: " Theoretical Mechanics. Statics and the Dynamics 
of a Particle," McGraw-Hill Book Company, Inc., New York, 1932. 

S. L. Loney: "Dynamics of a Particle and of Rigid Bodies," Cambridge 
University Press, Cambridge, 1923.. 

J. H. Jeans: "Theoretical Mechanics," Ginn and Company, Boston, 1907. 

E. T. Whittaker: "Analytical Dynamics," Cambridge University Press, 
Cambridge, 1928. 

R. C. Tolman: "Statistical Mechanics with Applications to Physics and 
Chemistry," Chemical Catalog Company, Inc., New York, 1927, Chap. II, 
The Elements of Classical Mechanics. 

W. E. Byerly: "Generalized Coordinates," Ginn and Company, Boston, 
1916. 



CHAPTER II 

THE OLD QUANTUM THEORY 

5. THE ORIGIN OF THE OLD QUANTUM THEORY 

The old quantum theory was born in 1900, when Max Planck 1 
announced his theoretical derivation of th& .distribution. law for, 
black-body radiation which he had previously formulated from 
empirical considerations. He showed that the results of experi- 
ment on the distribution of energy with frequency of radiation 
in equilibrium with matter at a given temperature can be 
accounted for by postulating that the vibrating particles of 
matter (considered to act as harmonic oscillators) do not emit 
or absorb light continuously but instead only in discrete quanti- 
ties of magnitude hv proportional to the frequency v of the light. 
The constant of proportionality, h, is a new constant of nature; 
it is called Planck 1 s constant and has the magnitude 6.547 X 10~ 27 
erg sec. Its dimensions (energy X time) are those of the old 
dynamical quantity called action; they are such that the product 
of h and frequency v (with dimensions sec -1 ) has the dimensions 
of energy. The dimensions of h are also those of angular momen- 
tum, and we shall see later that just as hv is a quantum of radiant 
energy of frequency v y so is h/2ic a natural unit or quantum of 
angular momentum. 

The development of the quantum theory was at first slow. It 
was not until 1905 that Einstein 2 suggested that the quantity 
of radiant energy hv was sent out in the process of emission of 
light not in all directions but instead unidirectionally, like a 
particle. The name light quantum or photon is applied to such a 
portion of radiant energy. Einstein also discussed the photo- 
electric effect, the fundamental processes of photochemistry, 
and the heat capacities of solid bodies in terms of 'the quantum 
theory. When light falls on a metal plate, electrons are emitted 
from it. The maximum speed of these photoelectrons, however, 

1 M. Planck, Ann. d. Phys. (4) 4, 553 (1901). 

2 A. Einstein, Ann. d. Phys. (4) 17, 132 (1905). 

25 



26 THE OLD QUANTUM THEORY II-5a 

is not dependent on the intensity of the light, as would be 
expected from classical electromagnetic theory, but only on its 
frequency; Einstein pointed out that this is to be expected from 
the quantum theory, the process of photoelectric emission involv- 
ing the conversion of the energy hv of one photon into the kinetic 
energy of a photoelectron (plus the energy required to remove 
the electron from the metal). Similarly, Einstein's law of 
photochemical equivalence states that one molecule may be 
activated to chemical reaction by the absorption of one photon. 

The third application, to the heat capacities of solid bodies, 
marked the beginning of the quantum theory of material systems. 
Planck's postulate regarding the emission and absorption of 
radiation in quanta hv suggested that a dynamical system such 
as an atom oscillating about an equilibrium position with fre- 
quency vo might not be able to oscillate with arbitrary energy, 
but only with energy values which differ from one another by 
integral multiples of hv . From this assumption and a simple 
extension of the principles of statistical mechanics it can be 
shown that the heat capacity of a solid aggregate of particles 
should not remain constant with decreasing temperature, but 
should at some low temperature fall off rapidly toward zero. 
This prediction of Einstein, supported by the earlier experi- 
mental work of Dewar on diamond, was immediately verified 
by the experiments of Nernst and Eucken on various substances; 
and quantitative agreement between theory and experiment for 
simple crystals was achieved through Debye's brilliant refinement 
of the theory. 1 

6a. The Postulates of Bohr. — The quantum theory had 
developed to this stage before it became possible to apply it 
to the hydrogen atom; for it was not until 1911 that there 
occurred the discovery by Rutherford of the nuclear constitu- 
tion of the atom — its composition from a small heavy posi- 
tively charged nucleus and one or more extranuclear electrons. 
Attempts were made immediately to apply the quantum theory to 
the hydrogen atom. The successful effort of Bohr 2 in 1913, 
despite its simplicity, may well be considered the greatest single 
step in the development of the theory of atomic structure. 

1 P. Debyb, Ann. d. Phya. (4) 39, 789 (1912); see also M. Born and T. votf 
KXrmAn, Phya. Z. 13, 297 (1912); 14, 15 (1913). 

S XT T>^ CT -, DL.'I %* A« t /*r\*n\ 



H-5a] THE ORIGIN OF THE OLD QUANTUM THEORY 27 

It was clearly evident that the laws of classical mechanical and 
electromagnetic theory could not apply to the Rutherford 
hydrogen atom. According to classical theory the electron 
in a hydrogen atom, attracted toward the nucleus by an inverse- 
square Coulomb force, would describe an elliptical or circular 
orbit about it, similar to that of the earth about the sun. ^The 
a ccelerati o n of the ch ar^edjgartjcles would lead to the emission 
of light, with frequencies equal to the mechanical frequency 
of the electron in its orbit, and to multiples of this as overtones. 
With the emission of energy, the radius of the orbit_ would 
diminish and the mechanical frequency would change. Hence 
the emitted light should show a wide range of frequencies.) This 
is not at all what is observed — the radiation emitted by hydrogen 
atoms is confined to spectral lines of sharply defined frequencies, 
and, moreover, these frequencies are not related to one another 
by integral factors, as overtones, but instead show an interesting 
additive relation, expressed in the Ritz combination principle, and 
in addition a still more striking relation involving the squares 
of integers, discovered by Balmer. Furthermore, the existence 
of stable non-radiating atoms was not to be understood on the 
basis of classical theory, for a system consisting of electrons 
revolving about atomic nuclei would be expected to emit radiant 
energy until the electrons had fallen into the nuclei. 

Bohr, no doubt inspired by the work of Einstein mentioned 
above, formulated the two following postulates, which to a great 
extent retain their validity in the quantum mechanics. 

I. The Existence of Stationary States. An atomic system can 
exist in certain stationary states f each one corresponding to a 
definite value of the energy W of the system; and transition from 
one stationary state to another is accompanied by the emission 
or absorption as radiant energy, or the transfer to or from 
another system, of an amount of energy equal to the difference 
in energy of the two states. 

II. The Bohr Frequency Rule. The frequency of the radiation 
emitted by a system on transition from an initial state of energy 
W2 to a final state of lower energy Wi (or absorbed on transition 
from the state of energy Wi to that of energy TF 2 ) is given by 
the equation 1 

1 This relation was suggested by the Ritz combination principle t which it 
closely resembles. It was found empirically by Rite and others that if 



28 THE OLD QUANTUM THEORY [H-5b 

W 2 ~ Wi 



v = 



(5-1) 



Bohr in addition gave a method of determining the quantized 
states of motion — the stationary states — of the hydrogen atom. 
His method of quantization, involving the restriction of the 
angular momentum of circular orbits to integral multiples of 
the quantum h/2ir, though leading to satisfactory energy 
levels, was soon superseded by a more powerful method, described 
in the next section. 

Problem 5-1. Consider an electron moving in a circular orbit about a 
nucleus of charge Ze. Show that when the centrifugal force is just balanced 
by the centripetal force Ze 2 /r 2 , the total energy is equal to one-half the 
potential energy —Ze 2 /r. Evaluate the energy of the stationary states for 
which the angular momentum equals nh/2ir } with n - 1, 2, 3, • • • . 

6b. The Wilson-Sommerfeld Rules of Quantization. — In 
1915 W. Wilson and A. Sommerfeld discovered independently 1 
a powerful method of quantization, which was soon applied, 
especially by Sommerfeld and his coworkers, in the discussion 

lines of frequencies v\ and vi occur in the spectrum of a given atom it is 
frequently possible to find also a line with frequency vi -f- ^ or v\ — j> 2 . 
This led directly to the idea that a set of numbers, called term values, can 
be assigned to an atom, such that the frequencies of all the spectral lines 
can be expressed as differences of pairs of term values. Term values are 
usually given in wave numbers, since this unit, which is the reciprocal 
of the wave length expressed in centimeters, is a convenient one for spectro- 
scopic use. We shall use the symbol v for term values in wave numbers, 
reserving the simpler symbol v for frequencies in sec -1 . The normal state 
of the ionized atom is usually chosen as the arbitrary zero, and the term 
values which represent states of the atom with lower energy than the ion 
are given the positive sign, so that the relation between W and v is 

~ = -- 
he 

The modern student, to whom the Bohr frequency rule has become common- 
place, might consider that this rule is clearly evident in the work of Planck 
and Einstein. This is not so, however; the confusing identity of the 
mechanical frequencies of the harmonic oscillator (the only system discussed) 
and the frequency of the radiation absorbed and emitted by this quantized 
system delayed recognition of the fact that a fundamental violation of 
electromagnetic theory was imperative. 

W. Wilson, Phil. Mag. 29, 795 (1915); A. Sommerfeld, Ann. d. Phys. 
51, 1 (1916). 



H-5c] THE ORIGIN OF THE OLD QUANTUM THEORY 29 

of the fine structure of the spectra of hydrogen and ionized 
helium, their Zeeman and Stark effects, and many other phe- 
nomena. The first step of their method consists in solving the 
classical equations of motion in the Hamiltonian form (Sec. 2), 
therefore making use of the coordinates £1, • • • , <?3n and the 
canonically conjugate momenta p h • • • , p 3n as the independent 
variables. The assumption is then introduced that only those 
classical orbits are allowed as stationary states for which the 
following conditions are satisfied: 

fpkdq k = n k h, k = 1, 2, • • • , 3n; n k = an integer. (5-2) 

These integrals, which are called action integrals, can be calcu- 
lated only for conditionally periodic systems; that is, for systems 
for which coordinates can be found each of which goes through a 
cycle as a function of the time, independently of the others. 
The definite integral indicated by the symbol f is taken over 
one cycle of the motion. Sometimes the coordinates can be 
chosen in several different ways, in which case the shapes of the 
quantized orbits depend on the choice of coordinate systems, but 
the energy values do not. 

We shall illustrate the application of this postulate to the 
determination of the energy levels of certain specific problems in 
Sections 6 and 7. 

5c. Selection Rules. The Correspondence Principle. — The 
old quantum theory did not provide a satisfactory method of cal- 
culating the intensities of spectral lines emitted or absorbed by 
a system, that is, the probabilities of transition from one sta- 
tionary state to another with the emission or absorption of a 
photon. Qualitative information was provided, however, by an 
auxiliary postulate, known as Bohr's correspondence principle, 
which correlated the quantum-theory transition probabilities 
with the intensity of the light of various frequencies which would 
have been radiated by the system according to classical electro- 
magnetic theory. In particular, if no light of frequency cor- 
responding to a given transition would have been emitted 
classically, it was assumed that the transition would not take 
place. The results of such considerations were expressed in 
selection rules. 

For example, the energy values nhv of a harmonic oscillator 
(as given in the following section) are such as apparently to 



30 THE OLD QUANTUM THEORY [H-6a 

permit the emission or absorption of light of frequencies which 
are arbitrary multiples (n 2 — ni)p of the fundamental fre- 
quency vq. But a classical harmonic oscillator would emit only 
the fundamental frequency vo f with no overtones, as discussed 
in Section 3 ; consequently, in accordance with the correspondence 
principle, it was assumed that the selection rule An = ± 1 was 
valid, the quantized oscillator being thus restricted to transitions 
to the adjacent stationary states. 

6. THE QUANTIZATION OF SIMPLE SYSTEMS 

6a. The Harmonic Oscillator. Degenerate States. — It was 

shown in the previous chapter that for a system consisting of 
a particle of mass m bound to the equilibrium position x — 
by a restoring force —kx= —4n 2 mv\x and constrained to move 
along the x axis the classical motion consists in a harmonic oscilla- 
tion with frequency v , as described by the equation 

x = x sin 2-Kvd. (6-1) 

The momentum p x = mx has the value 

p z = 2^771^0^0 cos 2wv t, (6-2) 

so that the quantum integral can be evaluated at once: 

(bpxdx = f '°m(27rvoa;o cos 2irvot) 2 dt = 2ir 2 v mxl = nh. (6-3) 

The amplitude x is hence restricted to the quantized values 
x 0n == {nh/2jr 2 v m}tt. The corresponding energy values are 

W n = T +V = 2r 2 m^xJ m (sin 2 2irv t + cos 2 2irv Q t) = 2r 2 mj>lx 2 0n , 
or 

W n = nhvo, n = 0, 1, 2, - • • . (6-4) 

Thus we see that the energy levels allowed by the old quantum 
theory are integral multiples of hv , as indicated in Figure 6-1. 
The selection rule An = ± 1 permits the emission and absorption 
of light of frequency v only. 

A particle bound to an equilibrium position in a plane by 
restoring forces with different force constants in the x and y 
directions, corresponding to the potential function 

V = Wm(vlx 2 + v 2 y 2 ) f (fr-5) 



n-6b] 



THE QUANTIZATION OF SIMPLE SYSTEMS 



31 



is similarly found to carry out independent harmonic oscillations 
along the two axes. The quantization restricts the energy to 
the values 



W nx ny = n x hv x + KlyhVy, 71*, Uy = 0, 1, 2, 

determined by the two quantum numbers n x and n v 



■ , (6-6) 

The ampli- 
tudes of motion x and y are given by two equations similar to 
Equation 6-3. 



/,wi 



v,w 





n-5 


V 


n-4 / 


n» 3 / 


\ n " 2 / 


\ n- \ / 


V / 



A ~^ 

Fig. 6-1. — Potential-energy function and quantized energy levels for the har- 
monic oscillator according to the old quantum theory. 

In case that v x = v v = v , the oscillator is said to be isotropic. 
The energy levels are then given by the equation 

W n = (n x + riy)hvo = nhv . (6-7) 

Different states of motion, corresponding to different sets of values 
of the two quantum numbers n 9 and n v , may then correspond 
to the same energy level. Such an energy level is said to be 
degenerate, the degree of degeneracy being given by the number 
of independent sets of quantum numbers. In this case the nth 
level shows (n + l)-fold degeneracy. The nth level of the 
three-dimensional isotropic harmonic oscillator shows 

- — --fold degeneracy. 

6b. The Rigid Rotator. — The configuration of the system of 
a rigid rotator restricted to a plane is determined by a single 
angular coordinate, say x« The canonically conjugate angular 
momentum, p x = Ix, where / is the moment of inertia, 1 is a 

1 See Section 36a. footnote, for a definition of moment of inertia. 



32 THE OLD QUANTUM THEORY [II-6c 

constant of the motion. 1 Hence the quantum rule is 

f Q *Pxdx = 2rrp x = Kh 
or 

Vx = g, K = 0, 1, 2, • • • . (6-8) 

Thus the angular momentum is an integral multiple of h/2ir, as 
originally assumed by Bohr. The allowed energy values are 

v 2 K 2 h 2 



W 



K=5 

The rigid rotator in space can be 

described by polar coordinates of 
the figure axis, <p and #. On apply- 
ing the quantum rules it is found 
K c 4 that the total angular momentum is 
given by Equation 6-8, and the 
component of angular momentum 
along the z axis by 

P, = ^ M=-K,-K + l, 

K-2 • • 7r - ,0, • • • , +K. (6-10) 



K«3 



1 ■■ ■- K=i The energy levels are given by 

"fig. G~2.~Energy levels iortL Equation 6-9, each level being 
rotator according to the old (2K + l)-fold degenerate, inas- 

quantum theory. much ag the quantum numbe r M 

does not affect the energy (Fig. 6-2). 

6c. The Oscillating and Rotating Diatomic Molecule. — A 
molecule consisting of two atoms bonded together by forces 
which hold them near to the distance r apart may be approxi- 
mately considered as a harmonic oscillator joined with a rigid 
rotator of moment of inertia / = nrf, ju being the reduced mass. 
The quantized energy levels are then given by the equation 

K 2 h 2 
W vK = ^o+J^j> (6-11) 

v being the oscillational or vibrational quantum number 2 and K 

1 Section le, footnote. 

* The symbol v is now used by band spectroscopists rather than n for this 
quantum number. 



H-6d] THE QUANTIZATION OF SIMPLE SYSTEMS 33 

the rotational quantum number. The selection rules for such a 
molecule involving two unlike atoms are AK = ±1, Av = ±1. 
Actual molecules show larger values of Au, resulting from devi- 
ation cf the potential function from that corresponding to 
harmonic oscillation. 

The frequency of light absorbed in a transition from the state 
with quantum numbers v" ', K" to that with quantum numbers 
t/, K' is 

v*>k».,'k' = (v f - v")v +(K'>- K">)~, 

or, introducing the selection rule AK = ±1, 

r,»K».*K»+i = {V - v")"o + (±2K" + 1)~ (6-12) 

The lines corresponding to this equation are shown in Figure 6-3 
for the fundamental oscillational band v = — > v = 1, together 

Calculated by equation 6~I2 

I I I I I I I I I I I I I I I I I I 

K)*? ?*88*7 7*6 6*5.5*44<*3 3-2 2*1 1*00*1 1*22*3 3*44-5 5*6 6-77*8 
Observed 



10*9 9*8 8*7 7*6 6*5 5*44*3 3*22*1 1*0 0*1 1*22*3 4*5 6*7 8*9 

v ^ 3*4 5*6 7*8 9*0 

Fig. d-3. — The observed rotational fine structure of the hydrogen chloride 
fundamental oscillational band v = — > v = 1, showing deviation from the 
equidistant spacing of Equation 6-12. 

with th3 experimentally observed absorption band for hydrogen 
chloride. It is seen that there is rough agreement; the observed 
lines are not equally spaced, however, indicating that our theo- 
retical treatment, with its assumption of constancy of the moment 
of inertia J, is too strongly idealized. 

6d. The Particle in a Box. — Let us consider a particle of mass 
m in a box in the shape of a rectangular parallelepiped with 
edges a, 6, and c, the particle being under the influence of no 



34 THE OLD QUANTUM THEORY [H-6e 

forces except during collision with the walls of the box, from 
which it rebounds elastically. The linear momenta p x , p v , and 
p t will then be constants of the motion, except that they will 
change sign on collision of the particle with the corresponding 
walls. Their values are restricted by the rule for quantization 
as follows: 

(ppxdx = 2ap x = n x h f p x = -~i n x = 0, 1, 2, • • • 

P* = 2jr> n, = 0, 1, 2, • • • 

Consequently the total energy is restricted to the values 

W^. = i(rf + rf + rf) - ^ + =| + J*)- («W4) 

6e. Diffraction by a Crystal Lattice. — Let us consider an 
infinite crystal lattice, involving a sequence of identical planes 
spaced with the regular interval d. The allowed states of motion 
of this crystal along the z axis we assume, in accordance with 
the rules of the old quantum theory, to be those for which 

jfpgdz = n z h. 

For this crystal it is seen that a cycle for the coordinate z is the 
identity distance d, so that (p z being constant in the absence of 
forces acting on the crystal) the quantum rule becomes 

* d n t h 

pjlz = nji, or p t = -~ (&-15) 



I 



Any interaction with another system must be such as to leave p 8 

quantized; that is, to change it by the amount Ap t = Artgh/d 

or nh/d, in which n = An* is an integer. One such type of 

interaction is collision with a photon of frequency v> represented 

in Figure 6-4 as impinging at the angle # and being specularly 

reflected. Since the momentum of a photon is hv/c, and its 

hv 
component along the z axis — sin #, the momentum transferred 

c 

to the crystal is — sin # = ~ sin #. Equating this with the 

C A 



II-6e] 



THE QUANTIZATION OF SIMPLE SYSTEMS 



35 



allowed momentum change of the crystal nh/d, we obtain the 
expression 

nX = 2d sin #. (6-16) 

This is, however, just the Bragg equation for the diffraction of 
x-rays by a crystal. This derivation from the corpuscular view 
of the nature of light was given 
by Duane and Compton 1 in 
1923. 

Let us now consider a particle, 
say an electron, of mass m simi- 
larly reflected by the crystal. 
The momentum transferred to 
the crystal will be 2mv sin #, 
which is equal to a quantum 
for the crystal when 

n— = 2d sin *. (6-17) 
mv 




6-4. — The reflection of a photon 
by a crystal. 



Fia. 



Thus we see that a particle would be scattered by a crystal only 
when a diffraction equation similar to the Bragg equation for 
x-rays is satisfied. The wave length of light is replaced by the 
expression 

h 



X = 



mv 



(6-18) 



which is indeed the de Broglie expression for the wave length 
associated with an electron moving with the speed v. This 
simple consideration, which might have led to the discovery of 
the wave character of material particles in the days when the 
old quantum theory had not yet been discarded, was overlooked 
at that time. 

In the above treatment, which is analogous to the Bragg treat- 
ment of x-ray diffraction, the assumption of specular reflection is 
made. This can be avoided by a treatment similar to Laue's 
derivation of his diffraction equations. 

The foregoing considerations provide a simple though perhaps 
somewhat extreme illustration of the power of the old quantum 
theory as well as of its indefinite character. That a formal argu- 
ment of this type leading to diffraction equations usually derived 

1 W. Duane, Proc. Nat. Acad. Sci. 9, 158 (1923); A. H. Compton, ibid. 
9, 359 U923). 



36 THE OLD QUANTUM THEORY [H-7a 

by the discussion of interference and reinforcement of waves 
could be carried through from the corpuscular viewpoint with the 
old quantum theory, and that a similar treatment could be given 
the scattering of electrons by a crystal, with the introduction of 
the de Broglie wave length for the electron, indicates that the 
gap between the old quantum theory and the new wave mechanics 
is not so wide as has been customarily assumed. The indefinite- 
ness of the old quantum theory arose from its incompleteness — 
its inability to deal with any systems except multiply-periodic 
ones. Thus in this diffraction problem we are able to derive 
only the simple diffraction equation for an infinite crystal, the 
interesting questions of the width of the diffracted beam, the dis- 
tribution of intensity in different diffraction maxima, the effect 
of finite size of the crystal, etc., being left unanswered. 1 

7. THE HYDROGEN ATOM 

The system composed of a nucleus and one electron, whose 
treatment underlies any theoretical discussion of the electronic 
structure of atoms and molecules, was the subject of Bohr's first 
paper on the quantum theory. 2 In this paper he discussed cir- 
cular orbits of the planetary electron about a fixed nucleus. 
Later 3 he took account of the motion of the nucleus as well as 
the electron about their center of mass and showed that with 
the consequent introduction of the reduced mass of the two 
particles a small numerical deviation from a simple relation 
between the spectral frequencies of hydrogen and ionized helium 
is satisfactorily explained. Sommerfeld 4 then applied his more 
general rules for quantization, leading to quantized elliptical 
orbits with definite spatial orientations, and showed that the 
relativistic change in mass of the electron causes a splitting of 
energy levels correlated with the observed fine structure of 
hydrogenlike spectra. In this section we shall reproduce the 
Sommerfeld treatment, except for the consideration of the rela- 
tivistic correction. 

7a. Solution of the Equations of Motion. — The system con- 
sists of two particles, the heavy nucleus, with mass m\ and 

1 The application of the correspondence principle to this problem was made 
by P. S. Epstein and P. Ehrenfest, Proc. Nat. Acad. Sci. 10, 133 (1924). 
* N. Bohr, Phil. Mag. 26, 1 (1913). 
8 N. Bohr, ibid. 27, 506 (1914). 
4 A. Sommerfeld, Ann. d. Phys. 51, 1 (1916). 



H-7a] THE HYDROGEN ATOM 37 

electric charge +Ze, and the electron, with mass m 2 and charge 
— e, between which there is operative an inverse-square attrac- 
tive force corresponding to the potential-energy function 

V(r) = -Ze 2 /r, 

r being the distance between the two particles. (The gravi- 
tational attraction is negligibly small relative to the electro- 
static attraction.) The system is similar to that of the sun and a 
planet, or the earth and moon. It was solved by Sir Isaac 
Newton in his " Philosophiae Naturalis Principia Mathematical 
wherein he showed that the orbits of one particle relative to the 
other are conic sections. Of these we shall discuss only the 
closed orbits, elliptical or circular, inasmuch as the old quantum 
theory was incapable of dealing with the hyperbolic orbits of the 
ionized hydrogen atom. 

The system may be described by means of Cartesian coordi- 
nates X\ 9 2/1, Z\ and x 2 , 2/2, z 2 °f the two particles. As shown in 
Section 2d by the introduction of coordinates x, y, z of the center 
of mass and of polar coordinates r, #, <p of the electron relative 
to the nucleus, the center of mass of the system undergoes 
translational motion in a fixed direction with constant speed, 
like a single particle in field-free space, and the relative motion 

of electron and nucleus is that of a particle of mass u = \ > 

nti + m a 

the reduced mass of the two particles, about a fixed center to 

which it is attracted by the same force as that between the 

electron, and nucleus. Moreover, the orbit representing any 

state of motion lies in a plane (Sec. Id). 

In terms of variables r and x in the plane of motion, the 

Lagrangian equations of motion are 

Ze 2 
pr = fxrx 2 - -p- (7-1) 

and 

j t (nr*x) = 0. (7-2) 

The second of these can be integrated at once (as in Sec. Id), to 
give 

ixr 2 x = Vi a constant. (7-3) 

This first result expresses Keoler's area law: The radius vector 



38 THE OLD QUANTUM THEORY [H-7a 

from sun to planet sweeps out equal areas in equal times. The 
constant p is the total angular momentum of the system. 
Eliminating x horn Equations 7-1 and 7-3, we obtain 

* r - £3 ~ IT' (7-4) 

which on multiplication by r and integration leads to 

The constant of integration W is the total energy of the system 
(aside from the translational energy of the system as a whole). 
Instead of solving this directly, let us eliminate t to obtain an 
equation involving r and x- Since 

. _ dr _ dr dx _ dr p 
~~ dt ~~ dxdt ~~ dx ^ 

Equation 7-5 reduces to 



(7-6) 



A dr_Y = _1 
\r 2 dx) r 2 



p'r 
or, introducing the new variable 



, 2Ze 2 n 2fiW 

2 + ^2 r "t- — * 



u = -; (7-7) 



r 



±d X = , d " (7-8) 



p" +"^ W 



This can be integrated at once, for W either positive or negative. 
In the latter case (closed orbits) there is obtained 

1 ZeV , 1 /4 M 2 Z 2 e 4 7 SnW . , . ,- a , 

«=;=-^r+2V~^- + '^ sm(x ~ Xo) - (7 " 9) 

This is the equation of an ellipse with the origin at one focus, as 
in Figure 7-1. In terms of the eccentricity e and the semimajor 
and semiminor axes a and 6, the equation of such an ellipse is 

r a(l - « 2 ) " ^ + 6 2 S n (x Xo) ' 

(7-10) 

with b = aVl - « 2 . 



n-7b] 



THE HYDROGEN ATOM 



39 



Thus it is found that the elements of the elliptical orbit are given 
by the equations 

™ h - P 1 - .2 _ J*E£ 

2W' " -v/Z^w ~~ M^V 



a = — 



(7-11) 



The energy TF is determined by the major axis of the ellipse 
alone. 

As shown in Problem 5-1, the total energy for a circular orbit 
is equal to one-half the potential energy and to the kinetic energy 
with changed sign. It can be shown also that similar relations 




*~* 



Fig. 7-1. 



-An elliptical electron-orbit for the hydrogen atom according to the 
old quantum theory. 



hold for the time-average values of these quantities for elliptic 
orbits, that is, that 

W = V 2 V = -T, (7-12) 

in which the barred symbols indicate the time-average values of 
the dynamical quantities. 

7b. Application of the Quantum Rules. The Energy Levels. — 
The Wilson-Sommerfeld quantum rules, in terms of the polar 
coordinates r, #, and <p } are expressed by the three equations 

tfprdr = n r h, (7-13a) 

fp*d& = n*h, (7-136) 

ffydv = mh. (7-13c) 

Since p? is a constant (Sec. le) f the third of these can be integrated 
at once, giving 



40 THE OLD QUANTUM THEORY P-7b 

2*p v = mh, or p,-g, m=±l, ±2, • • • . 

(7-14) 

Hence the component of angular momentum of the orbit along the 
z axis can assume only the quantized values which are integral 
multiples of h/2ir. The quantum number m is called the mag- 
netic quantum number, because it serves to distinguish the 
various slightly separated levels into which the field-free energy 
levels are split upon the application of a magnetic field to the 
atom. This quantum number is closely connected with the 
orientation of the old-quantum-theory orbit in space, a question 
discussed in Section Id. 

The second integral is easily discussed by the introduction of 
the angle x and its conjugate momentum p x = p, the total 
angular momentum of the system, by means of the relation, 
given in Equation 1-41, Section le, 

Vxdx = P*d& + Pedcp. (7-15) 

In this way we obtain the equation 

f Vx d x = kh, (7-16) 

in which p x is a constant of the motion and k is the sum of n& 
and m. This integrates at once to 

%rp = kh, or p = £"*, k = 1, 2, • • • . (7-17) 

Hence the total angular momentum of the orbit was restricted 
by the old quantum theory to values which are integral mul- 
tiples of the quantum unit of angular momentum h/2w. The 
quantum number k is called the azimuthal quantum number. 

To evaluate the first integral it is convenient to transform it 
in the following way, involving the introduction of the angle % 
and the variable u = 1/r with the use of Equation 7-6: 

f^-^-K$ dx '*•$$**• (7 - 18) 

From Equation 7-10 we find on differentiation 



du _ e cos (x - xo) 
d x o(l - « 2 ) ' 



(7-19) 



II-7b] THE HYDROGEN ATOM 41 

with the use of which the r quantum condition reduces to the 
form 

pe 2 p cos2 .(x - xo) d = nX (y^) 

Jo {1 + €sm( x - xo)} 2 

The definite integral was evaluated by Sommerfeld. 1 The 
resultant equation is 

^(vf=, - l ) - nA (7 " 21) 

This, with the value of p of Equation 7-17 and the relation 
b = a\/l — e 2 , leads to the equation 

_ _ ___ _ _. ( 7- 22 ) 

In this equation we have introduced a new quantum number n> 
called the total quantum number , as the sum of the azimuthal 
quantum number k and the radial quantum number n r : 

n = n r + k. (7-23) 

With these equations and Equation 7-11, the energy values 
of the quantized orbits and the values of the major and minor 
semiaxes can be expressed in terms of the quantum numbers 
and the physical constants involved. The energy is seen to 
have the value 

Wn = -**«* . _*«*, (7-24) 

n 2 h 2 n 2 f v 7 

being a function of the total quantum number alone. The value 
of jffi, the Rydberg constant^ which is given by the equation 

R . *£. (7-25) 

depends on the reduced mass \x of the electron and the nucleus. 
It is known very accurately, being obtained directly from 
spectroscopic data, the values as reported by Birge for hydrogen, 
ionized helium, and infinite nuclear mass being 

R n = 109,677.759 ± 0.05 cm" 1 , 
R He = 109,722.403 ± 0.05 cm- 1 , 
fl w = 109,737.42 ± 0.06 cm" 1 . 

1 A. Sommerfeld, Ann. d. Phys. 51, 1 (1916). 



42 THE OLD QUANTUM THEORY [H-7b 

The major and minor semiaxes have the values 

n 2 a . nka n . 

a = -£-) o = —=-) (7-26) 



«« - OT (7-27) 



in which the constant a has the value 

The value of this quantity, which for hydrogen is the distance of 
the electron from the nucleus in the circular orbit with n = 1, 
k = 1, also depends on the reduced mass, but within the experi- 
mental error in the determination of e the three cases mentioned 
above lead to the same value 1 

a = 0.52851, 

in which 1 A = 1 X 10~ 8 cm. The energy may also be expressed 
in terms of a as 

Ze 2 Z 2 e 2 

The total energy required to remove the electron from the 
normal hydrogen atom to infinity is hence 

W» = ^ = iW. - £ (7-29) 

This quantity, TT H = 2.1528 X 10"" 11 ergs, is often expressed in 
volt electrons, TP H = 13.530 v.e., or in reciprocal centimeters or 
wave numbers, Wn = 109,677.76 cm"" 1 (the factor he being 
omitted), or in calories per mole, Wn = 311,934 cal/mole. 

The energy levels of hydrogen are shown in Figure 7-2. It is 
seen that the first excitation energy, the energy required to raise 
the hydrogen atom from the normal state, with n = 1, to the 
first excited state, with n = 2, is very large, amounting to 
10.15 v.e. or 234,000 cal/mole. The spectral lines emitted by 
an excited hydrogen atom as it falls from one stationary state to 
another would have wave numbers or reciprocal wave lengths v 
given by the equation 

p = R -(w^ ~ ^)' (7 " 30) 

1 The value given by Birge for infinite mass is 

0.5281 e* ± 0.0004 X 10" f cm, 
that for hydrogen being 0.0003 larger (Appendix I). 



H-7c] 



THE HYDROGEN ATOM 



43 



in which n" and n' are the values of the total quantum number 
for the lower and the upper state, respectively. The series of 
lines corresponding to n" = 1, that is, to transitions to the normal 
state, is called the Lyman series, and those corresponding to 
n" = 2, 3, and 4 are called the Balmer, Paschen, and Brackett 
series, respectively. The Lyman series lies in the ultraviolet 
region, the lower members of the Balmer series are in the visible 
region, and the other series all lie in the infrared. 



w-o 



W«-Rhc 




n-1 ■ 



Fig. 7-2.- 



Lyman 
series 

—The energy levels of the hydrogen atom, and the transitions giving 
rise to the Lyman, Balmer, Paschen, and Brackett series. 



7c. Description of the Orbits. — Although the allowed orbits 
given by the treatment of Section 76 are not retained in the 
quantum-mechanical model of hydrogen, they nevertheless 
serve as a valuable starting point for the study of the more subtle 
concepts of the newer theories. The old-quantum-theory orbits 
are unsatisfactory chiefly because they restrict the motion too 
rigidly, a criticism which is generally applicable to the results of 
this theory. 

For the simple non-relativistic model of the hydrogen atom in 
field-free space the allowed orbits are certain ellipses whose com- 
mon focus is the center of mass of the nucleus and the electron, 
and whose dimensions are certain functions of the quantum 



44 



THE OLD QUANTUM THEORY 



[II-7c 



numbers, as we have seen. For a given energy level o! the 
atom there is in general more than one allowed ellipse, since the 
energy depends only on the major axis of the ellipse and not on 
its eccentricity or orientation in space. These different ellipses 
are distinguished by having different values of the azimuthal 





k-1 





Fig. 7-3a, b, c- 



-Bohr-Sommerfeld electron-orbits for n 
to the same scale. 



1, 2, and 3, drawn 



quantum number fc, which may be any integer from 1 to n. 
WTienJ^quidsji^ the orbit is a circle, as is seen from Equation 
7-26. For k less than n, the minor semiaxis b is less than the 
major semiaxis a, the eccentricity e of the orbit increasing as k 
decreases relative to n. The value zero for k was somewhat 
arbitrarily excluded, on the basis of the argument that the 



n-7d] THE HYDROGEN ATOM 45 

corresponding orbit is a degenerate line ellipse which would 
cause the electron to strike the nucleus. 

Figure 7-3 shows the orbits for n = 1, 2, and 3 and for the 
allowed values of A;. The three different ellipses with n = 3 
have major axes of the same length and minor axes which 
decrease with decreasing A;. Figure 7-3 also illustrates the 
expansion of the orbits with increasing quantum number, the 
radii of the circular orbits increasing as the square of n. 

A property of these orbits which is of particular importance in 
dealing with heavier atoms is the distance of closest approach of 
the electron to the nucleus. Using the expressions for a and b 
given in Equation 7-26 and the properties of the ellipse, we obtain 

71 ( 71 —— ■ "\/ 71 —~~ Jc )fLn 

for this distance the value — ~ — This formula 

and the orbits drawn in Figure 7-3 show that the most eccentric 
orbit for a given n, i.e., that with the smallest-value. of fc, comes 
the nearest to the nucleus. In many-electron atoms, this 
causes a separation of the energies corresponding to these 
different elliptical orbits with the same n, since the presence of 
the other electrons, especially the inner or core electrons, causes a 
modification of the field acting on the electron when it enters 
the region near the nucleus. 

Since the charge on the nucleus enters the expression for the 
radius of the orbit given by Equations 7-26 and 7-27, the orbits 
for He + are smaller than the corresponding ones for hydrogen, 
the major semiaxis being reduced one-half by the greater charge 
on the helium-ion nucleus. 

7d. Spatial Quantization. — So far we have said nothing of 
the orientation of the orbits in space. If a weak field, either 
electric or magnetic, is applied to the atom, so that the z direction 
in space can be distinguished but no appreciable change in 
energy occurs, the z component of the angular momentum of 
the atom must be an integral multiple of h/2ir, as mentioned 
in Section 76 following Equation 7-14. This condition, which 
restricts the orientation of the plane of the orbit to certain definite 
directions, is called spatial quantization. The vector representing 
the total angular momentum p is a line perpendicular to the 
plane of the orbit (see Sec. le) and from Equation 7-17 has the 
length kh/2r. The z component of the angular momentum is 
of length fc cos <a(h/%c) 9 if w is the angle between the vector p and 



46 THE OLD QUANTUM THEORY [H-7d 

the z axis. This results in the following expression for cos o>: 

m 
cos co = -r • 

The value zero for m was excluded for reasons related to those 
used in barring k = 0, so that m may be ±1, ±2, • • • , ±k. 



m-+2 




(m»0) 



m»-3 



Fig. 7-4a, b, c. — Spatial quantization of Bohr-Sommerfeld orbits with A; = 1, 2. 

and 3. 

For the lowest state of hydrogen, in which k = 1 (and for all orbits 
for which k = 1), there are only two values of m, +1 and — 1, 
which correspond to motion in the xy plane in a counterclockwise 
or in a clockwise sense. For fc = 2 four orientations are per- 



D-8] THE DECLINE OF THE OLD QUANTUM THEORY 47 

mitted, as shown in Figure 7-4. Values ±k for m always cor- 
respond to orbits lying in the xy plane. 

It can be shown by the methods of classical electromagnetic 
theory that the motion of an electron with charge — e and mass 

kh 
m in an orbit with angular momentum ^- gives rise to a magnetic 

kh 6 
field corresponding to a magnetic dipole of magnitude ^-^ — 

oriented in the same direction as the angular momentum vector. 

The component of magnetic moment in the direction of the z axis 

he 

is m- A The energy of magnetic interaction of the atom with 

47rra c 

he 
a magnetic field of strength H parallel to the- z axis is mj- — H. 

It was this interaction energy which was considered to give rise 

to the Zeeman effect (the splitting of spectral lines by a magnetic 

field) and the phenomenon of paramagnetism. It is now known 

that this explanation is only partially satisfactory, inasmuch as 

the magnetic moment associated with the spin of the electron, 

discussed in Chapter VIII, also makes an important contribution. 

he 
The magnetic moment j- — I s called a Bohr magneton. 

Problem 7-1. Calculate the frequencies and wave lengths of the first 
five members of the Balmer series for the isotopic hydrogen atom whose 
mass is approximately 2.0136 on the atomic weight scale, and compare with 
those for ordinary hydrogen. 

Problem 7-2. Quantize the system consisting of two neutral particles 
of masses equal to those of the electron and proton held together by gravita- 
tional attraction, obtaining expressions for the axes of the orbits and the 
energy levels. 

8. THE DECLINE OF THE OLD QUANTUM THEORY 

The historical development of atomic and molecular mechanics 
up to the present may be summarized by the following division 
into periods (which, of course, are not so sharply demarcated as 
indicated) : 

1913-1920. The origin and extensive application of the old 

quantum theory of the atom. 
1920-1925. The decline of the old quantum theory. 
1925- . The origin of the new quantum mechanics and 
its application to physical problems. 



48 THE OLD QUANTUM THEORY [H-8 

1927- The application of the new quantum mechanics 

to chemical problems. 
The present time may well be also the first part of the era of the 
development of a more fundamental quantum mechanics, includ- 
ing the theory of relativity and of the electromagnetic field, and 
dealing with the mechanics of the atomic nucleus as well as of the 
extranuclear structure. 

vThe decline of the old quantum theory began with the introduc- 
tion of half-integral values for quantum numbers in place of 
integral values for certain systems, in order to obtain agreement 
with experiment* It was discovered that the pure rotation 
spectra of the hydrogen halide molecules are not in accordance 
with Equation 6-9 with K = 0, 1, 2, • • • , but instead require 
K = }i, %, • • • . Similarly, half -integral values of the oscilla- 
tional quantum number v in Equation 6-11 were found to be 
required in order to account for the observed isotope displace- 
ments for diatomic molecules. Half-integral values for the 
azimuthal quantum number k were also indicated by observations 
on both polarization and penetration of the atom core by a 
valence electron. Still more serious were cases in which agree- 
ment with the observed energy levels could not be obtained by 
the methods of the old quantum theory by any such subterfuge 
or arbitrary procedure (such as the normal state of the helium 
atom, excited states of the helium atom, the normal state of 
the hydrogen molecule ion, etc.), and cases where the methods 
of the old quantum theory led to definite qualitative disagreement 
with experiment (the influence of a magnetic field on the dielectric 
constant of a gas, etc.). Moreover, the failure of the old quan- 
tum theory to provide a method of calculating transition probabil- 
ities and the intensities of spectral lines was recognized more 
and more clearly as a fundamental flaw. Closely related to this 
was the lack of a treatment of the phenomenon of the disper- 
sion of light, a problem which attracted a great amount of 
attention. 

This dissatisfaction with the old quantum theory culminated 
in the formulation by Heisenberg 1 in 1925 of his quantum 
mechanics, as a method of treatment of atomic systems leading 
to values of the intensities as well as frequencies of spectral 
lines. The quantum mechanics of Heisenberg was rapidly 

1 W. Heisenberg, Z. /. Phys. 33, 879 (1925). 



H-8] THE DECLINE OF THE OLD QUANTUM THEORY 49 

developed by Heisenberg, Born, and Jordan 1 by the introduction 
of matrix methods. In the meantime Schrodinger had inde- 
pendently discovered and developed his wave mechanics, 2 
stimulated by the earlier attribution of a wave character to the 
electron by de Broglie 3 in 1924. The mathematical identity of 
matrix mechanics and wave mechanics was then shown by 
Schrodinger 4 and by Eckart. 5 The further development of the 
quantum mechanics was rapid, especially because of the con- 
tributions of Dirac, who formulated 6 a relativistic theory of the 
electron and contributed to the generalization of the quantum 
mechanics (Chap. XV). 

General References on the Old Quantum Theory 

A. Sommerfeld: " Atomic Structure and Spectral Lines," E. P. Dutton & 
Co., Inc., New York, 1923. 

A. E. Ruark and H: C. Urey: "Atoms, Molecules and Quanta," McGraw- 
Hill Book Company, Inc., New York, 1930. 

1 M. Born and P. Jordan, ibid. 34, 858 (1925); M. Born, W. Heisenberg, 
and P. Jordan, ibid. 35, 557 (1926). 

2 E. Schrodinger, Ann. d. Phys. 79, 361, 489; 80, 437; 81, 109 (1926). 

3 L. de Broglie, Thesis, Paris, 1924; Ann. de phys. (10) 3, 22 (1925). 

4 E. Schrodinger, Ann. d. Phys. 79, 734 (1926). 
6 C. Eckart, Phys. Rev. 28, 711 (1926). 

•P. A. M. Dirac, Proc. Roy. Soc. A 113, 621; 114, 243 (1927); 117, 610 
(1928). 



CHAPTER III 

THE SCHRODINGER WAVE EQUATION WITH THE 
HARMONIC OSCILLATOR AS AN EXAMPLE 

In the preceding chapters we have given a brief discussion of 
the development of the theory of mechanics before the discovery 
of the quantum mechanics. Now we begin the study of the quan- 
tum mechanics itself, starting in this chapter with the Schrodinger 
wave equation for a system with only one degree of freedom, the 
general principles of the theory being illustrated by the special 
example of the harmonic oscillator, which is treated in great 
detail because of its importance in many physical problems. 
The theory will then be generalized in the succeeding chapter 
to systems of point particles in three-dimensional space. 

9. THE SCHR6DINGER WAVE EQUATION 
In the first paragraph of his paper 1 Quantisierung als Eigen- 
wertproblem, communicated to the Annalen der Physik on 
January 27, 1926, Erwin Schrodinger stated essentially: 

In this communication I wish to show, first for the simplest case of 
the non-relativistic and unperturbed hydrogen atom, that the usual 
rules of quantization can be replaced by another postulate, in which 
there occurs no mention of whole numbers. Instead, the introduction 
of integers arises in the same natural way as, for example, in a vibrating 
string, for which the number of nodes is integral. The new conception 
can be generalized, and I believe that it penetrates deeply into the true 
nature of the quantum rules. 

In this and four other papers, published during the first half of 
1926, Schrodinger communicated his wave equation and applied 
it to a number of problems, including the hydrogen atom, the 
harmonic oscillator, the rigid rotator, the diatomic molecule, and 

1 E. SchrOdingbr, Ann. d. Phys. 79, 361 (1926), and later papers referred 
to on the preceding page. An English translation of these papers has 
appeared under the title E. Schrodinger, "Collected Papers on Wave 
Mechanics," Blackie and Son, London and Glasgow, 1928. 

60 



ra-9] the schrOdinger WAVE EQUATION 51 

the hydrogen atom in an electric field (Stark effect). For the 
last problem he developed his perturbation theory, and for 
the discussion of dispersion he also developed the theory of a 
perturbation varying with the time. His methods were rapidly 
adopted by other investigators, and applied with such success 
that there is hardly a field of physics or chemistry that has 
remained untouched by Schrodinger's work. 

Schrodinger's system of dynamics differs from that of Newton, 
Lagrange, and Hamilton in its aim as well as its method. Instead 
of attempting to find equations, such as Newton's equations, 
which enable a prediction to be made of the exact positions and 
velocities of the particles of a system in a given state of motion, 
he devised a method of calculating a function of the coordinates 
of the system and the time (and not the momenta or velocities), 
with the aid of which, in accordance with the interpretation 
developed by Born, 1 probable values of- the coordinates and 
of other dynamical quantities can be predicted for the system. 
It was later recognized that the acceptance of dynamical equa- 
tions of this type involves the renunciation of the hope of describ- 
ing in exact detail the behavior of a system. The degree of 
accuracy with which the behavior of a system can be discussed 
by quantum-mechanical methods forms the subject of Heisen- 
berg y s uncertainty 'principle? to which we shall recur in Chapter 
XV. 

The Schrodinger wave equation and its auxiliary postulates 
enable us to determine certain functions ^ of the coordinates of a 
system and the time. These functions are called the Schrodinger 
wave junctions or probability amplitude functions. The square 
of the absolute value of a given wave function is interpreted as 
a probability distribution function for the coordinates of the 
system in the state represented by this wave function, as will 
be discussed in Section 10a. The wave equation has been 
given this name because it is a differential equation of the second 
order in the coordinates of the system, somewhat similar to the 
wave equation of classical theory. The similarity is not close, 
however, and we shall not utilize the analogy in our exposition. 

Besides yielding the probability amplitude or wave function ty f 
the Schrodinger equation provides a method of calculating values 

1 M. Born, Z. f. Phys. 37, 863; 38, 803 (1926). 
* W. Heisenberg, Z. /. Phys. 43, 172 (1927). 



52 THE SCHRODINGER WAVE EQUATION [HI-9 

of the energy of the stationary states of a system, the existence 
of which we have discussed in connection with the old quantum 
theory. No arbitrary postulates concerning quantum numbers 
are required in this calculation; instead, integers enter auto- 
matically in the process of finding satisfactory solutions of the 
wave equation. 

For our purposes, the Schrodinger equation, the auxiliary 
restrictions upon the wave function % and the interpretation of 
the wave function are conveniently taken as fundamental 
postulates, with no derivation from other principles necessary. 

This idea may be clarified by a comparison with other branches 
of physics. Every department of deductive science must 
necessarily be founded on certain postulates which are regarded 
as fundamental. Frequently these fundamental postulates are 
so closely related to experiment that their acceptance follows 
directly upon the acceptance- of the experiments upon which 
they are based, as, for example, the inverse-square law of electrical 
attraction. In other cases the primary postulates are not so 
directly obvious from experiment, but owe their acceptance to the 
fact that conclusions drawn from them, often by long chains of 
reasoning, agree with experiment in all of the tests which have 
been made. The second law of thermodynamics is representative 
of this type of postulate. It is not customary to attempt to 
derive the second law for general systems from anything more 
fundamental, nor is it obvious that it follows directly from 
some simple experiment; nevertheless, it is accepted as correct 
because deductions made from it agree with experiment. It is 
an assumption, justified only by the success achieved by its 
consequences. 

The wave equation of Schrodinger belongs to this latter class 
of primary assumption. It is not derived from other physical 
laws nor obtained as a necessary consequence of any experiment; 
instead, it is assumed to be correct, and then results predicted 
by it are compared with data from the laboratory. 

A clear distinction must frequently be made between the way 
in which a discoverer arrives at a given hypothesis and the 
logical position which this hypothesis occupies in the theory when 
it has been completed and made orderly and deductive. In 
the process of discovery, analogy often plays a very important 
part. Thus the analogies between geometrical optics and 



m-9a] THE SCHRODINGER WAVE EQUATION 53 

classical mechanics on the one hand and undulatory optics and 
wave mechanics on the other may have assisted Schrodinger to 
formulate his now famous equation; but these analogies by no 
means provide a logical derivation of the equation. 

In many cases there is more than one way of stating the funda- 
mental postulates. Thus either Lagrange's or Hamilton's form 
of the equations of motion may be regarded as fundamental for 
classical mechanics, and if one is so chosen, the other can be 
derived from it. Similarly, there are other ways of expressing 
the basic assumptions of quantum mechanics, and if they are 
used, the wave equation can be derived from them, but, no 
matter which mode of presenting the theory is adopted, some 
starting point must be chosen, consisting of a set of assumptions 
not deduced from any deeper principles. 

It often happens that principles which have served as the basis 
for whole branches of theory are superseded by other principles 
of wider applicability. Newton's laws of motion, adopted 
because they were successful in predicting the motions of the 
planets and in correlating celestial and terrestrial phenomena, 
were replaced by Lagrange's and Hamilton's equations because 
these are more general. They include Newton's laws as a 
special case and in addition serve for the treatment of motions 
involving electric, magnetic, and relativistic phenomena. Like- 
wise, quantum mechanics includes Newton's laws for the special 
case of heavy bodies and in addition is successful in problems 
involving atoms and electrons. A still more general theory 
than that of Schrodinger has been developed (we shall discuss 
it in Chap. XV), but for nearly all purposes the wave equation is a 
convenient and sufficient starting point. 

9a. The Wave Equation Including the Time. — Let us first 
consider a Newtonian system with one degree of freedom, 
consisting of a particle of mass m restricted to motion along a 
fixed straight line, which we take as the x axis, and let us assume 
that the system is further described by a potential-energy func- 
tion V (x) throughout the region — <x> < x < + «> . For this 
system the Schrodinger wave equation is assumed to be 



8r 2 m dx 2 ' w v ' ' 2iri dt 

In this equation the function ^(z, t) is called the Schrddinger 



54 THE SCHRODINGER WAVE EQUATION [m-9a 

wave function including the time, or the probability amplitude 
function. It will be noticed that the equation is somewhat 
similar in form to the wave equations occurring in other branches 
of theoretical physics, as in the discussion of the motion of a 
vibrating string. The student facile in mathematical physics 
may well profit from investigating this similarity and also the 
analogy between classical mechanics and geometrical optics on 
the one hand, and wave mechanics and undulatory optics on the 
other. 1 However, it is not necessary to do this. An extensive 
previous knowledge of partial differential equations and their 
usual applications in mathematical physics is not a necessary 
prerequisite for the study of wave mechanics, and indeed the 
study of wave mechanics may provide a satisfactory introduction 
to the subject for the more physically minded or chemically 
minded student. 

The Schrodinger time equation is closely related to the equation 
of classical Newtonian mechanics 

H(p„ x) = T(p.) + V(z) = W, (9-2) 

which states that the total energy W is equal to the sum of the 
kinetic energy T and the potential energy V and hence to the 
Hamiltonian function H(p x , x). Introducing the coordinate x 
and momentum p x , this equation becomes 

ff(Px, x) = JLp» + V(x) = W. (9-3) 

If we now arbitrarily replace p x by the differential operator 

h A h A 

s— . — and W by — s-r -zv and introduce the function ¥(x, t) on 
2m dx Van dt 

which these operators can operate, this equation becomes 



H (hh x ) 



* ( *> () - "ass w + v * - -33 aT (9_4) 



which is identical with Equation 9-1. The wave equation is 

1 See, for example, Condon and Morse, "Quantum Mechanics," p. 
10, McGraw-Hill Book Company, Inc., New York, 1929; Ruark and Urey, 
"Atoms, Molecules and Quanta," Chap. XV, McGraw-Hill Book Company, 
Inc., New York, 1930; E. Schr6dinger, Ann. d. Phys. 79, 489 (1926); K. K. 
Darrow, Rev. Mod. Phys. 6, 23 (1934); or other treatises on wave mechanics, 
listed at the end of this chapter. 



m-9a] THE SCHRODINOER WAVE EQUATION 55 

consequently often conveniently written as 

H* « W*, (9-5) 

h A l a 

in which it is understood that the operators s— -r- and — tt— -^ 
are to be introduced. 

Iji replacing p x by the operator — ; — > p* is to be replaced by ( — -. ) — > 
* 2m dx \2m/ dx* 

and so on. (In some cases, which, however, do not arise in the simpler 
problems which we are discussing in this book, there may be ambiguity 
regarding the formulation of the operator. 1 ) It might be desirable to dis- 
tinguish between the classical Hamiltonian function H = H{p Xt x) and the 
Hamiltonian operator 



H 



\2m dx / 



as by writing /^operator for the latter. We shall not do this, however, since 

the danger of confusion is small. Whenever H is followed by ^ (or by ^, 

representing the wave functions not including the time, discussed in the 

following sections), it is understood to be the Hamiltonian operator. Simi- 

h d 

larly, whenever W is followed by ^ it represents the operator : — 

2t% dt 

The symbol W will also be used to represent the energy constant (Sees. 

96, 9c). We shall, indeed, usually restrict the symbol W to this use, and 

h * , . 

wnte — for the operator. 

2iri dt 

It must be recognized that this correlation of the wave equation 
and the classical energy equation, as well as the utilization 
which we shall subsequently make of many other classical 
dynamical expressions, has only formal significance. It provides 
a convenient way of describing the system for which we are 
setting up a wave equation by making use of the terminology 
developed over a long period of years by the workers in classical 
dynamics. Thus our store of direct knowledge regarding the 
nature of the system known as the hydrogen atom consists in the 
results of a large number of experiments — spectroscopic, chemical, 
etc. It is found that all of the known facts about this system 
can be correlated and systematized (and, we say, explained) 
by associating with this system a certain wave equation. Our 
confidence in the significance of this association increases when 
predictions regarding previously uninvestigated properties of 

1 B. Podolsky, Phys. Rev. 32, 812 (1928). 



56 THE SCHR6DINGER WAVE EQUATION [III-9b 

the hydrogen atom are subsequently verified by experiment. 
We might then describe the hydrogen atom by giving its wave 
equation; this description would be complete. It is unsatis- 
factory, however, because it is unwieldy. On observing that 
there is a formal relation between this wave equation and the 
classical energy equation for a system of two particles of different 
masses and electrical charges, we seize on this as providing a 
simple, easy, and familiar way of describing the system, and 
we say that the hydrogen atom consists of two particles, the 
electron and proton, which attract each other according to 
Coulomb's inverse-square law. Actually we do not know that 
the electron and proton attract each other in the same way 
that two macroscopic electrically charged bodies do, inasmuch 
as the force between the two particles in a hydrogen atom has 
never been directly measured. All that we do know is that the 
wave equation for the hydrogen atom bears a certain formal 
relation to the classical dynamical equations for a system of 
two particles attracting each other in this way. 

Having emphasized the formal nature of this correlation and 
of the usual description of wave-mechanical systems in terms of 
classical concepts, let us now point out the extreme practical 
importance of this procedure. It is found that satisfactory wave 
equations can be formulated for nearly all atomic and molecular 
systems by accepting the descriptions of them developed during 
the days of the classical and old quantum theory and translating 
them into quantum-mechanical language by the methods 
discussed above. Indeed, in many cases the wave-mechanical 
expressions for values of experimentally observable properties of 
systems are identical with those given by the old quantum theory, 
and in other cases only small changes are necessary. Throughout 
the following chapters we shall make use of such locutions as 
"a system of two particles with inverse-square attraction" 
instead of "a system whose wave equation involves six coordi- 
nates and a function e 2 / r i2," etc. 

9b. The Amplitude Equation. — In order to solve Equation 9-1, 
let us (as is usual in the solution of a partial differential equation 
of this type) first study the solutions ^ (if any exist) which can 
be expressed as the product of two functions, one involving the 
time alone and the other the coordinate alone : 

¥(*, t) * HxMt). 



m-9b] THE SCHRODINGER WAVE EQUATION 57 

On introducing this in Equation 9-1 and dividing through by 
4s(x)<p(t), it becomes 

The right side of this equation is a function of the time t alone 
and the left side a function of the coordinate x alone. It is 
consequently necessary that the value of the quantity to which 
each side is equal be dependent on neither x nor t; that is, that 
it be a constant. Let us call it W. Equation 9-6 can then 
be written as two equations, namely, 

_ = -—wxo 

and \ (9-7) 

The second of these is customarily written in the form 

2 2 + 8 -p{W- 7(*))*-0, (9-8) 

obtained on multiplying by — 87r 2 ra//i 2 and transposing the term 
in IF. 

Equation 9-8 is often itself called the Schrodinger wave equa- 
tion, or sometimes the amplitude equation, inasmuch as \l/(x) 
determines the amplitude of the function >£(#, t). It is found 
that the equation possesses various satisfactory solutions, cor- 
responding to various values of the constant W. Let us indicate 
these values of W by attaching the subscript n, and similarly 
represent the amplitude function corresponding to W n as \p n (x). 
The corresponding equation for <p(t) can be integrated at once 

to give 

w 

The general solution of Equation 9-1 is the sum of all the particu- 
lar solutions with arbitrary coefficients. We consequently 
write as the general expression for the wave function for this 
system 

*(*, t) = %an*n(x, t) - 2a n * n (s)e~ 2 ""A (9-10) 



-w^ (9-9) 



58 THE SCHRODINGER WAVE EQUATION [m-9c 

in which the quantities a n are constants. The symbol V is 

n 

to be considered as representing the process of summation over 
discrete values of W n or integration over a continuous range or 
both, according to the requirements of the particular case. 

It will be shown later that the general postulates which we 
shall make regarding the physical interpretation of the wave 
function require that the constant W n represent the energy of 
the system in its various stationary states. 

9c. Wave Functions. Discrete and Continuous Sets of 
Characteristic Energy Values. — The functions ^W which 
satisfy Equation 9-8 and also certain auxiliary conditions, dis- 
cussed below, are variously called wave functions or eigenf unctions 
(Eigenfunktionen), or sometimes amplitude functions, charac- 
teristic functions, or proper functions. It is found that satis- 
factory solutions \l/ n of the wave equation exist only for certain 
values of the parameter W n (which is interpreted as the energy 
of the system). These values W n are characteristic energy values 
or eigenvalues (Eigenwerte) of the wave equation. A wave 
equation of this type is called a characteristic value equation. 

Inasmuch as we are going to interpret the square of the absolute 
value of a wave function as having the physical significance of a 
probability distribution function, it is not unreasonable that the 
wave function be required to possess certain properties, such as 
single-valuedness, necessary in order that this interpretation be 
possible and unambiguous. It has been found that a satisfactory 
wave mechanics can be constructed on the basis of the following 
auxiliary postulates regarding the nature of wave functions: 

To be a satisfactory wave function, a solution of the Schrodinger 
wave equation must be continuous, single-valued, and finite 1 through- 

1 The assumption that the wave function be finite at all points in configura- 
tion space may be more rigorous than necessary. Several alternative 
postulates have been suggested by various investigators. Perhaps the most 
satisfying of these is due to W. Pauli ("Handbuch der Physik," 2d ed., Vol. 
XXVI, Part 1, p. 123). In Section 10 we shall interpret the function ^*^ as a 
probability distribution function. In order that this interpretation may be 
made, it is necessary that the integral of ^*SIr over configuration space be a 
constant with changing time. Pauli has shown that this condition is satis- 
fied provided that ¥ is finite throughout configuration space, but that it is 
also satisfied in certain cases by functions which are not finite everywhere. 
The exceptional cases are rare and do not occur in the problems treated iD 
this book. 



m-9c] 



THE SCHRODINGER WAVE EQUATION 



59 



out the configuration space of the system (that is, for all values of 
the coordinate x which the system can assume). 

These conditions are those usually applied in mathematical 
physics to functions representing physical quantities. For 
example, the function representing the displacement of a vibrat- 
ing string from its equilibrium configuration would have to 
satisfy them. 

For a given system the characteristic energy values W n may 
occur only as a set of discrete values, or as a set of values covering 
a continuous range, or as both. From analogy with spectroscopy 
it is often said that in these three cases the energy values comprise 
a discrete spectrum, a continuous spectrum, or both. The way 





w 3 




\v Wj 


X 


W, 






w x>v 


• W ° V 


•x-a 



Fia. 9-1. — Potential-energy function for a general system with one degree of 

freedom. 

in which the above postulates regarding the wave equation and 
its acceptable solutions lead to the selection of definite energy 
values may be understood by the qualitative consideration of a 
simple example. Let us consider, for our system of one degree 
of freedom, that the potential-energy function V(x) has the form 
given in Figure 9-1, such that for very large positive or negative 
values of x, V(x) increases without limit. For a given value of 
the energy parameter W, the wave equation is 



dH 



-*8vm 



W}+. 



(9-11) 



dx 2 h 2 

In the region of large x (x > a) the quantity V(x) - W will be 
positive. Hence in this region the curvature ^ wil1 be positive 
if $ is positive, and negative if \p is negative. Now let us assume 



60 



THE SCHRODINQER WAVE EQUATION 



[in-9c 



that at an arbitrary point x = c the function \j/ has a certain value 
(which may be chosen arbitrarily, inasmuch as the wave equation 

is a homogeneous equation 1 ) and a certain slope -p> as indicated 

for Curve 1 in Figure 9-2. The behavior of the function, as it 
is continued both to the right and to the left, is completely 
determined by the values assigned to two quantities; to wit, the 

slope -j- at the point x = c, and the energy parameter W in the 

wave equation, which determines the value of the second deriva- 




Fig. 9-2. — The behavior of \p for x > a. 



tive. As we have drawn Curve 1, the curvature is determined 
by the wave equation to be negative in the region x < a, where 
V(x) — W is negative, \p being positive, and hence the curve can 
be continued to the right as shown. At the point x = a, the 
function remaining positive, the curvature becomes positive, the 
curve then being concave upward. If the slope becomes positive, 
as indicated, then the curve will increase without limit for 
increasing x, and as a result of this "infinity catastrophe'' the 
function will not be an acceptable wave function. 

1 An equation is homogeneous in \j/, if the same power of \J/ (in our case the 
first power) occurs in every term. The function obtained by multiplying 
any solution of a homogeneous equation by a constant is also a solution 
of the equation. 



III-9c] THE SCHRODINGER WAVE EQUATION 61 

We can now make a second attempt, choosing the slope at 
x = c as indicated for Curve 3. In this case the curve as drawn 
intersects the x axis at a point x = d to the right of a. For 
values of x larger than d the function \p is negative, and the curva- 
ture is negative. The function decreases in value more and more 
rapidly with increasing x, again suffering the infinity catastrophe, 
and hence it too is not an acceptable wave function in this 
region. 

Thus we see that, for a given value of W, only by a very careful 
selection of the slope of the function at the point x = c can the 
function be made to behave properly for large values of x. This 
selection, indicated by Curve 2, is such as to cause the wave 
function to approach the value zero asymptotically with increas- 
ing x. 

Supposing that we have in this way determined, for a given 
value of W y a value of the slope at x = c which causes the 
function to behave properly for large positive values of x, we 
extend the function to the left and consider its behavior for large 
negative values of x. In view of our experience on the right, 
it will not be surprising if our curve on extension to the left 
behaves as Curve 1 or Curve 3 on the right, eliminating the 
function from consideration; in fact, it is this behavior which 
is expected for an arbitrarily chosen value of W. We can now 
select another value of W for trial, and determine for it the value 
of the slope at x = c necessary to cause the function to behave 
properly on the right, and then see if, for it, the curve behaves 
properly on the left also. Finally, by a very careful choice of 
the value of the energy parameter W, we are able to choose a 
slope at x = c which causes the function to behave properly 
both for very large and for very small values of x. This value 
of W is one of the characteristic values of the energy of the 
system. In view of the sensitiveness of the curve to the param- 
eter W, an infinitesimal change from this satisfactory value will 
cause the function to behave improperly. 

We conclude that the parameter W and the slope at the point 
x = c (for a given value of the function itself at this point) can 
have only certain values if \p is to be an acceptable wave function. 
For each satisfactory value of W there is one (or, in certain 
cases discussed later, more than one) satisfactory value of the 
slope, by the use of which the corresponding wave function can 



62 



THE SCHRODINGER WAVE EQUATION 



[in-9c 



be built up. For this system the characteristic values W n 
of the energy form a discrete set, and only a discrete set, inasmuch 
as for every value of W, no matter how large, V{x) — W is 
positive for sufficiently large positive or negative values of x. 

It is customary to number the characteristic energy values for 
such a system as indicated in Figure 9-1, W being the lowest, 
W\ the next, and so on, corresponding to the wave functions 
y l / o(x) t ^i(z), etc. The integer n, which is written as a subscript 
in W n and ^„(a:), is called the quantum number. For such a 
one-dimensional system it is equal to the number of zeros 1 
possessed by \l/ n . A slight extension of the argument given above 



W 

orJi 
V' 




X-> 



Fia. 9-3. — The energy levels for a system with V( — <») or F(-}-oo) finite. 

shows that all of the zeros lie in the region between the points 
x = b and x = a, outside of which V(x) — W n remains positive. 
The natural and simple way in which integral quantum numbers 
are introduced and in which the energy is restricted to definite 
values contrasts sharply with the arbitrary and uncertain 
procedure of the old quantum theory. 

Let us now consider a system in which the potential-energy 
function remains finite at x •—► + °° or at x — > — oo or at both 
limits, as shown in Figure 9-3. For a value of W smaller than 
both F(+<») and F(—oo) the argument presented above is 
valid. Consequently the energy levels will form a discrete set 
for this region. If W is greater than F (+«>), however, a 
similar argument shows that the curvature will be such as always 
to return the wave function to the x axis, about which it will 

1 A zero of ^*(x) is a point {x = xi) at which \J/ n is equal to zero. 



m-lOa] PHYSICAL INTERPRETATION OF WAVE FUNCTIONS 63 

oscillate. Hence any value of W greater than F(+°°) or 
V{ — oo ) will be an allowed value, corresponding to an acceptable 
wave function, and the system will have a continuous spectrum 
of energy values in this region. 

9d. The Complex Conjugate Wave Function W*(x, t).—ln 
the physical interpretation of the wave equation and its solutions, 
as discussed in the following section, the quantity ^*(x, t), 
the complex conjugate of ^(x, t), enters on an equivalent basis 
with V(x, t). The wave equation satisfied by V* is the complex 
conjugate of Equation 9-1, namely, 

_ h* a*9*(x, t) v , (x w (x t) _± a**(*, (Q _ l2) 

8^i dx* +V W * {X ' l) - 2ri dt (9 12) 

The general solution of this conjugate wave equation is the 
following, the conjugate of 9-10: 

*•(*, = £<*:*„*(*, = XaMix)**^. (9-13) 

n n 

(Some authors have adopted the convention of representing 
by the symbol Mr the wave function which is the solution of 
Equation 9-12 and by ^* that of 9-1. This is only a matter of 
nomenclature.) 

It will be noticed that in the complex conjugate wave function 
the exponential terms containing the time are necessarily different 
from the corresponding terms in ^ itself, the minus sign being 
removed to form the complex conjugate. The amplitude 
functions $n(x) } on the other hand, are frequently real, in which 
case ^*(x) = fn(x). 

10. THE PHYSICAL INTERPRETATION OF THE WAVE FUNCTIONS 

10a. W*(x, t)V(x, t) as a Probability Distribution Function — 
Let us consider a given general solution ^(x, t) of the wave equa- 
tion. For a given value of the time t, the function V*(x, t)¥(x, t) y 
the product of ¥ and its complex conjugate, is a function defined 
for all values of x between — oo and + » ; that is, throughout the 
configuration space of this one-dimensional system. We now 
make the following postulate regarding the physical significance 
of *: 

The quantity V*(x, t)V(x, t)dx is the probability thai the system 
in the physical situation represented by the wave function V(x t t) 



64 THE SCHRODINOER WAVE EQUATION lIH-lOb 

have at the time t the configuration represented by a point in the 
region dx of configuration space. In other words, ty*(x, t)^(x, t) 
is a probability distribution junction for the configuration of the 
system. In the simple system under discussion, ^*(x, t)V(x, t)dx 
is the probability that the particle lie in the region between x 
and x + dx at the time t. 

In order that this postulate may be made, the wave function 
^{x y t) must be normalized to unity (or, briefly, normalized); 
that is, the constants a n of Equation 9-10 must be so chosen as 
to satisfy the relation 

* + "Vfo t)*(x, t)dx = 1, (10-1) 



/- 



inasmuch as the probability that the coordinate x of the particle 
lie somewhere between — oo and +oo is necessarily unity. 
It is also convenient to normalize the individual amplitude 
functions ^ n (x) to unity, so that each satisfies the equation 

* + °Vn* (x)Mx)dx = 1. (10-2) 



J- 



Moreover, as proved in Appendix III, it is found that the 
independent solutions of any amplitude equation can always be 
chosen in such a way that for any two of them, \l/ m (x) and yp n {x) } 
the integral $$m(x)yl/ n (x)dx over all of configuration space van- 
ishes; that is, 

f*y%(x)+ n (x)dx =0, m * n. (10-3) 

The functions are then said to be mutually orthogonal. Using 
these relations and Equations 9-10 and 9-13, it is found that a 

wave function ty(x, t) = Va n ^ n (a;, t) is normalized when the 

n 

coefficients a n satisfy the relation 

X a nX = 1. (10-4) 

n 

10b. Stationary States. — Let us consider the probability dis- 
tribution function ty*ty for a system in the state represented by 



the wave function ^(x y t) = ^a n \l/ n (x)e ** h and its conjugate 

n 

S£*(x, t) = 2^a*\l/*(x)e h On multiplying these series 



ni-lOc] PHYSICAL INTERPRETATION OF WAVE FUNCTIONS 65 
together, ^*^ is seen to have the form 






M (W.-W.) f 

A 



in which the prime on the double-summation symbol indicates 
that only terms with m ?± n are included. In general, then, 
the probability function and hence the properties of the system 
depend on the time, inasmuch as the time enters in the exponen- 
tial factors of the double sum. Only if the coefficients a n are 
zero for all except one value of W n is ^*^ independent of t. 
In such a case the wave function will contain only a single term 

(with n = n'j say) Sfv(x, t) = i/ n '(x)e h , the amplitude 
function i/v(z) being a particular solution of the amplitude 
equation. For such a state the properties of the system as given 
by the probability function ^*^ are independent of the time, and 
the state is called a stationary state. 

10c. Further Physical Interpretation. Average Values of 
Dynamical Quantities. — If we inquire as to what average value 
would be expected on measurement at a given time t of the 
coordinate x of the system in a physical situation represented by 
the wave function ^, the above interpretation of ^*^ leads to the 
answer 

•-f 00 

y*(x, t^ixy t)xdx\ 



f- 



that is, the value of x is averaged over all configurations, using 
the function SF*^ as a weight or probability function. A similar 
integral gives the average value predicted for x 2 , or x 3 , or any 
function F(x) of the coordinate x: 

p= r*"JV(3, t)*(x, t)F(x)dx. (10-5) 

In order that the same question can be answered for a more 
general dynamical function G(p x , x) involving the momentum p x 
as well as the coordinate x, we now make the following more 
general postulate: 

The average value of the dynamical function G(p Xf x) predicted 
for a system in the physical situation represented by the wave 
function V(x, t) is given by the integral 



66 



THE SCHRODINGER WAVE EQUATION 



[m-10c 



= f + "**(*, Og/^. ±, * W «)d«, (10-6) 

in which the operator (?, obtained from ©(p*, a:) by replacing p x 

h /J 

by 5—. — > operates on the function ^f{x } t) and the integration is 

extended throughout the configuration space of the system. 1 

In general, the result of a measurement of G will not be given 
by this expression for G. G rather is the average of a very 
large number of measurements made on a large number of 
identical systems in the physical situation represented by ^, or 
repeated on the same system, which before each measurement 
must be in the same physical situation. For example, if SF is 




Fig. 10-1. — Two types of probability distribution function ^e*9, 

finite for a range of values of x (Curve A } Figure 10-1), then a 
measurement of x might lead to any value within this range, 
the probability being given by SF*^. Only if ^*^ were zero 
for all values of x except x = a, as indicated by Curve B in 
Figure 10-1, would the probability of obtaining a particular 
value x = a on measurement of x be unity. In this case the 
value a r would be predicted with probability unity to be obtained 
on measurement of the rth power of x; so that for such a prob- 
ability distribution function x r is equal to (x) r . It has also 
been shown by mathematicians that the existence of this identity 
of G r and (G) r for all values of r is sufficient to establish that the 
probability distribution function for the dynamical quantity G is 
of type B) that is, that the value of G can be predicted accurately. 

1 In some cases further considerations are necessary in order to determine 
the exact form of the operator, but we shall not encounter such difficulties. 



m-lla] HARMONIC OSCILLATOR IN WAVE MECHANICS 67 

Even if the system is in a stationary state, represented by the 

wave function V n (x, t) = \l/ n (x)e h , only an average value 
can be predicted for an arbitrary dynamical quantity. The 
energy of the system, corresponding to the Hamiltonian function 
H(Px, x), has, however, a definite value for a stationary state 
of the system, equal to the characteristic value W n found on 
solution of the wave equation, so that the result of a measurement 
of the energy of the system in a given stationary state can be 
predicted accurately. To prove this, we evaluate H r and {B) r . 
H is given by the integral 

the factor involving the time being equal to unity. This trans- 
forms with the use of Equation 9-8 into 

^ = /J 1 " y:(x)Wn+n(x)dx, 

\l/*(x)\l/ n (x)dx = 1, 

H = W n , and (H) r - W n . (10-7) 

By a similar procedure, involving repeated use of Equation 9-8, 
it is seen that W is equal to W r n . We have thus shown H r to 
be equal to (H) r , in consequence of which, in accordance with the 
argument set forth above, the energy of the system has the 
definite value W n . 

Further discussion of the physical significance of wave functions 
will be given in connection with the treatment of the harmonic 
oscillator in this chapter and of other systems in succeeding 
chapters, and especially in Chapter XV, in which the question 
of deciding which wave function to associate with a given system 
under given circumstances will be treated. In the earlier sections 
we shall restrict the discussion mainly to the properties of 
stationary states. 

11. THE HARMONIC OSCILLATOR IN WAVE MECHANICS 

11a. Solution of the Wave Equation. — As our first example 
of the solution of the Schrodinger wave equation for a dynamical 
system we choose the one-dimensional harmonic oscillator, not 
only because this provides a good illustration of the methods 



68 THE SCHRODINGER WAVE EQUATION [Hl-lla 

employed in applying the wave equation, but also because this 
system is of considerable importance in applications which we 
shall discuss later, such as the calculation of the vibrational 
energies of molecules. The more difficult problem of the three- 
dimensional oscillator was treated by the methods of classical 
mechanics in Section la, while the simple one-dimensional 
case was discussed according to the old quantum theory in 
Section 6a. 

The potential energy may be written, as before, in the form 
V(x) = 2w 2 mv$x 2 , in which x is the displacement of the particle 
of mass m from its equilibrium position x = 0. Insertion of this 
in the general wave equation for a one-dimensional system 
(Eq. 9-8) gives the equation 

g + ^(W - arWoZ*)* = 0, (11-1) 

or, introducing for convenience the quantities X = 8w 2 mW/h 2 
and a = 47r 2 m^ //i, 

g + (X - «»**)* = 0. (11-2) 

We desire functions yp{x) which satisfy this equation throughout 
the region of values — <x> to + °° for x ) and which are acceptable 
wave functions, i.e., functions which are continuous, single- 
valued, and finite throughout the region. A straightforward 
method of solution which suggests itself is the use of a power- 
series expansion for ^, the coefficients of the successive powers 
of x being determined by substitution of the series for yp in the 
wave equation. There is, however, a very useful procedure 
which we may make use of in this and succeeding problems, 
consisting of the determination of the form of ^ in the regions of 
large positive or negative values of x } and the subsequent dis- 
cussion, by the introduction of a factor in the form of a power 
series (which later reduces to a polynomial), of the behavior of ^ 
for \x\ small. This procedure may be called the polynomial 
method. 1 

The first step is the asymptotic solution of the wave equation 
when | a; | is very large. For any value of the energy constant W, 
a value of |x| can be found such that for it and all larger values 

1 A. Sommbbfeld, "Wave Mechanics," p. 11. 



m-lla] HARMONIC OSCILLATOR IN WAVE MECHANICS 69 

of \x\, X is negligibly small relative to <xV, the asymptotic form 
of the wave equation thus becoming 

2* = «V*. (11-3) 

This equation is satisfied asymptotically by the exponential 
functions 

\b = P. 2 



inasmuch as the derivatives of \p have the values 

= ±<xxe 



#_ ^..^i* 



dx 
and 

•T-s = a 2 x 2 e 2 + ae 2 
ax 2 ~ 

and the second term in ^— 2 is negligible in the region considered. 

Of the two asymptotic solutions e 2 and e 2 } the second is 
unsatisfactory as a wave function since it tends rapidly to 
infinity with increasing values of \x\; the first, however, leads to a 
satisfactory treatment of the problem. 

We now proceed to obtain an accurate solution of the wave 
equation throughout configuration space (—«)<£<+«>), 
based upon the asymptotic solution, by introducing as a factor 
a power series in x and determining its coefficients by substitution 
in the wave equation. 

Let $ = e~* xt f(x). Then 

= e~-* Xi {a*x>f -a}- 2axf +/"}, 

in which /' and /" represent -p- and -r~> respectively. Equation 

1 1-2 then becomes, on division by e 2 , 

f> - 2axf + (X - a)} = 0, (11-4) 

the terms in a 2 x 2 f cancelling. 



70 THE SCHRODINGER WAVE EQUATION [m-lla 

It is now convenient to introduce a new variable { , related to 
x by the equation 

i = V**, (11-5) 

and to replace the' function f(x) by H(£), to which it is equal. 
The differential equation 11-4 then becomes 



2 *f + C- 1 ) H = - ^ 



dm 

dp 

We now represent H (£) as a power series, which we differentiate 
to obtain its derivatives, 

v 

^ = 2" ( " " 1)a ^'" 2 = 1 • 2 «2 + 2 • 3a 3 £ + • • • . 

v 

On substitution of these expressions, Equation 11-6 assumes the 
following form: 

1 • 2a 2 + 2 • 3a 3 £ + 3 • 4a 4 £ 2 + 4 • 5a 5 £ 3 + ■ • ■ 
- 2ai£ - 2 • 2a 2 £ 2 - 2 • 3a 3 £ 3 - • • • 

+0- i )- + (s- i )'" f+ C- i )'" £!+ 



a->) 



«,{» + • • • =0. 

In order for this series to vanish for all values of £ (i.e., for #(£) 
to be a solution of 11-6), the coefficients of individual powers of 
£ must vanish separately 1 : 

1 • 2oi + Q - 1 Ja„ = 0, 

2 • 3a 3 + Q - 1 - 2 V = 0, 

3 • 4a 4 + Q - 1 - 2 • 2 Ja 2 = 0, 

4 • 5a 6 + Q - 1 - 2 • 3 Ja 8 = 0, 
1 See footnote, Sec. 23. 



m-lla] HARMONIC OSCILLATOR IN WAVE MECHANICS 71 
or, in general, for the coefficient of $", 

(v + !)(* + 2K +2 + Q - 1 - 2vja y = 



or 



(=-»-) 



a ' +2 = -(„ + !)(„ + 2)°" (1I_7) 

This expression is called a recursion formula. It enables the 
coefficients a 2 , a 8 , a 4 , • • • to be calculated successively in 
terms of a and a\, which are arbitrary. If a is set equal to zero, 
only odd powers appear; with a± zero, the series contains even 
powers only. 

For arbitrary values of the energy parameter X, the above 
given series consists of an infinite number of terms and does 
not correspond to a satisfactory wave function, because, as we 
shall show, the value of the series increases too rapidly as x 
increases, with the result that the total function, even though it 
includes the negative exponential factor, increases without 
limit as x increases. To prove this we compare the series for 
H and that for e ft , 

£4 £6 tv tv+2 

eP = 1 + I 2 + h + h + • • • +tV + -7 s — n- + • • • ■ 



2! ' 3! 



©' (H< 



For large values of J the first terms of these series will be unim- 
portant. Suppose that the ratio of the coefficients of the vth 
terms in the expansion of H (J) and e? is called c, which may be 
small or large, i.e., a y /b y = c, if b y is the coefficient of £" in the 
expansion of e*\ For large enough values of v, we have the 
asymptotic relations 



so that 



a v4 .2 = —a>v and b v +i = —b P9 
v v 



CLy+2 0t V 

by+2 by 



if v is large enough. Therefore, the higher terms of the series for 
H differ from those for e** only by a multiplicative constant, so 
that for large values of |£|, for which the lower terms are unim- 



72 THE SCHRODINGER WAVE EQUATION [III-lla 

portant, H will behave like e? and the product e 2 H will behave 

like e 2 in this region, thus making it unacceptable as a wave 

function. 

We must therefore choose the values of the energy parameter 

which will cause the series for H to break off after a finite number 

of terms, leaving a polynomial. This yields a satisfactory wave 

_I 2 
function, because the negative exponential factor e 2 will cause 

the function to approach zero for large values of |£|. The value 

of X which causes the series to break off after the nth term is 

seen from Equation 11-7 to be 

X = (2n + l)a. (11-8) 

It is, moreover, also necessary that the value either of a or of a i 
be put equal to zero, according as n is odd or even, inasmuch as a 
suitably chosen value of X can cause either the even or the odd 
series to break off, but not both. The solutions are thus either 
odd or even functions of £. This condition is a sufficient condi- 
tion to insure that the wave equation 11-2 have satisfactory 
solutions, and it is furthermore a necessary condition; no other 
values of X lead to satisfactory solutions. For each integral 
value 0, 1, 2, 3, • • • of n, which we may call the quantum 
number of the corresponding state of the oscillator, a satisfactory 
solution of the wave equation will exist. The straightforward 
way in which the quantum number enters in the treatment of 
the wave equation, as the degree of the polynomial #(£), is 
especially satisfying when compared with the arbitrary assump- 
tion of integral or half-integral multiples of h for the phase 
integral of the old quantum theory. 

The condition expressed in Equation 11-8 for the existence 
of the nth wave function becomes 

W = W n = (n + H)hv , n = 0, 1, 2, • • • , (11-9) 

when X and a are replaced by the quantities they represent. A 
comparison with the result W = nhv obtained in Section 6a 
by the old quantum theory shows that the only difference is 
that all the energy levels are shifted upward, as shown in Figure 
11— 1, by an amount equal to half the separation of the energy 
levels, the so-called zero-point energy %hv . From this we 



Hi-lib] HARMONIC OSCILLATOR IN WAVE MECHANICS 73 

see that even in its lowest state the system has an energy greater 
than that which it would have if it were at rest in its equilibrium 
position. The existence of a zero-point energy, which leads to 
an improved agreement with experiment, is an important feature 
of the quantum mechanics and recurs in many problems. 1 
Just as in the old-quantum-theory treatment, the frequency 
emitted or absorbed by a transition between adjacent energy 
levels is equal to the classical vibration frequency v (Sec. 40c). 





V 


n-5 




n-4 / 


i I 


n-3 / 


VJY 


n-2 / 




n-1 / 




\ n«0 / 




V y 



Fig. 11-1. — Energy levels for the harmonic oscillator according to wave me* 
chanics (see Fig. 6-1). 

lib. The Wave Functions for the Harmonic Oscillator and 
Their Physical Interpretation. — For each of the characteristic 
values W n of the energy, a satisfactory solution of the wave 
equation 11-1 can be constructed by the use of the recursion 
formula 11-7. Energy levels such as these, to each of which 
there corresponds only one independent wave function, are said 
to be non-degenerate to distinguish them from degenerate energy 
levels (examples of which we shall consider later), to which several 

1 The name zero-point energy is used for the energy of a system in its lowest 
stationary state because the system in thermodynamic equilibrium with its 
environment at a temperature approaching the absolute zero would be in 
this stationary state. The zero-point energy is of considerable importance 
in many statistical-mechanical and thermodynamic discussions. The 
existence of zero-point energy is correlated with the uncertainty principle 
(Chap. XV), 



74 



THE SCHRODINOER WAVE EQUATION 



m-llb] 



independent wave functions correspond. The solutions of 11-1 
may be written in the form 

*.(*) =N n e~W n (S), (11-10) 

in which £ = \/ax. H n (£) is a polynomial of the nth degree in 
£ , and N n is a constant which is adjusted so that \p n is normalized, 
i.e., so that 4/ n satisfies the relation 



f*y:(x)M*)dx = 1, 



(11-H) 



in which \f/* 9 the complex conjugate of ^ n , is in this case equal to 
^ n . In the next section we shall discuss the nature and properties 

























































Vol&i 






























V 


















\ 









« ■ 


j -i 


> - 


( 


) 






\ 



[*M\ 



_zr 


1 I 


55 


3u 


CX 


-v ^t 


^7 ~E_ 



-1 



1 



-3 -2 

«-* *-* 

Fig. 11-2. — The wave function ^o(£) for the normal state of the harmonic 

oscillator (left), and the corresponding probability distribution function 

[^o(£)P (right). The classical distribution function for an oscillator with the 
same total energy is shown by the dashed curve. 

of these solutions \p n in great detail. The first of them, which 
corresponds to the state of lowest energy for the system, is 

**(*) = ($ e ~ 2 = (irf e ~**' (11 " 12) 

Figure 11-2 shows this function. From the postulate discussed 
in Section 10a, ^J^ = ^§, which is also plotted in Figure 11-2, 
represents the probability distribution function for the coordinate 
x. In other words, the quantity \[/l(x)dx at any point x gives 
the probability of finding the particle in the range dx at that 
point. We see from the figure that the result of quantum 
mechanics for this case does not agree at all with the probability 
function which is computed classically for a harmonic oscillator 
with the same energy. Classically the particle is most likely to 



m-llbj HARMONIC OSCILLATOR IN WAVE MECHANICS 75 



be found at the ends of its motion, which are clearly defined points 
(the classical probability distribution is shown by the dotted 
curve in Figure 11-2), whereas \j/l has its maximum at the origin 
of a: and, furthermore, shows a rapidly decreasing but nevertheless 
finite probability of finding the particle outside the region allowed 
classically. This surprising result, that it is possible for a 
particle to penetrate into a region in which its total energy is less 
than its potential energy, is closely connected with Heisenberg's 



nd 



/\ 


A 


£j 


- n.,f ^ 


^\ i- " 


V J 


\-t 


Xtr 





-4-3-2-101234 



























i 


(\ 






/N 


L 












1 






\ 










/ 




1 




\ 








/ 


t 




H 


^ 


\ 












\ 




















\ 




















\ 




















\ 




















t. 


"^ 








-4 


- 


! -', 


I ' 


c 


) 




i : 


5 i 


\ 

























A 


/ 


\ 












A 


/ 








/' 


J 


y 














i 




\ 




\ 






( 














/ 


1 












\ 


/ 


u 












\ 


/ 


V 








-£ 


\ ", 


? -; 


> -i 


r*i,„. 

1 ' 


I s 


\ 4 


i 



*a$ 











| 1 


















n«< 


» 
























































































































































*~ 








■A 




5 - 


2 - 


( 


) 




? 


3 ^ 


I 



-^ 


4t 


R B 


-n-n^ 


±s 


=*# 


£== 


it 

-4 -3 -2-1 


2 3 4 




-4-3-2-101234 

Fig. 11-3. — The wave functions ^ n (£), n — 1 to 6, for the harmonic oscillator. 
For each case the heavy horizontal line indicates the region traversed by the 
classical harmonic oscillator with the same total energy. 

uncertainty principle, which leads to the conclusion that it is 
not possible to measure exactly both the position and the velocity 
of a particle at the same time. We shall discuss this phenomenon 
further in Chapter XV. It may be mentioned at this point, 
however, that the extension of the probability distribution func- 
tion into the region of negative kinetic energy will not require 
that the law of the conservation of energy be abandoned. 

The form of \p n for larger values of n is shown in Figure 11-3. 
Since H n is a polynomial of degree n, \f/ n will, have n zeros or 
points where \p n crosses the zero line. The probability of finding 
the particle at these points is zero. Insoection of Figure 11-3 
shows that all the solutions plotted show a general behavior in 



76 



THE SCHRODINGER WAVE EQUATION 



[m-llb 



agreement with that obtained by the general arguments of Section 
9c; that is, inside the classically permitted region of motion of 
the particle (in which V{x) is less than W n ) the wave function 
oscillates, having n zeros, while outside that region the wave 
function falls rapidly to zero in an exponential manner and has 
no zeros. Furthermore, we see in this example an illustration 
of still another general principle: The larger the value of n, the 




Fio. 11-4. — The probability distribution function [^io(£)] 2 for the state 
n ■■ 10 of the harmonic oscillator. Note how closely the function approximates 
in its average value the probability distribution function for the classical har- 
monic oscillator with the same total energy, represented by the dashed curve. 

more nearly does the wave-mechanical probability distribution 

function approximate to the classical expression for a particle 

with the same energy. Figure 11-4 shows \l/ 2 (x) for the state 

with n = 10 compared with the classical probability curve for 

21 
the harmonic oscillator with the same value -yhv for the energy. 

It is seen that, aside from the rapid fluctuation of the wave- 
mechanical curve, the general agreement of the two functions 
is good. This agreement permits us to visualize the motion of 
the particle in a wave-mechanical harmonic oscillator as being 



m-llc] HARMONIC OSCILLATOR IN WAVE MECHANICS 77 

similar to its classical to-and-fro motion, the particle speeding 
up in the center of its orbit and slowing down as it approaches 
its maximum displacement from its equilibrium position. The 
amplitude of the oscillation cannot be considered to be constant, 
as for the classical oscillator ; instead, we may picture the particle 
as oscillating sometimes with very large amplitude, and some- 
times with very small amplitude, but usually with an amplitude 
in the neighborhood of the classical value for the same energy. 
Other properties of the oscillator also are compatible with this 
picture; thus the wave-mechanical root-mean-square value of the 
momentum is equal to the classical value (Prob. 11-4). 

A picture of this type, while useful in developing an intuitive 
feeling for the wave-mechanical equations, must not be taken 
too seriously, for it is not completely satisfactory. Thus it 
cannot be reconciled with the existence of zeros in the wave 
functions for the stationary states, corresponding to points where 
the probability distribution function becomes vanishingly 
small. 

lie. Mathematical Properties of the Harmonic Oscillator 
Wave Functions. — The polynomials H n (£) and the functions 
_v 
e 2 H n (£) obtained in the solution of the wave equation for the 

harmonic oscillator did not originate with Schrodinger's work 
but were well known to mathematicians in connection with other 
problems. Their properties have been intensively studied. 

For the present purpose, instead of developing the theory of 
the polynomials H n (£), called the Hermite polynomials, from the 
relation between successive coefficients given in Equation 11-7, 
it is more convenient to introduce them by means of another 
definition: 

We shall show later that this leads to the same functions as 
Equation 11-7. A third definition involves the use of a generating 
function, a method which is useful in many calculations and which 
is also applicable to other functions. The generating function 
for the Hermite polynomials is 

CO 

S(«, 8) m e«- <-«>' m 2^T 8B - (U-14) 



f8->0 



78 THE SCHR6DINQER WAVE EQUATION [Hl-lic 

This identity in the auxiliary variable s means that the function 
e*t- (•-{>* hag foe property that, if it is expanded in a power series 
in s, the coefficients of successive powers of s are just the Hermite 
polynomials H n (£), multiplied by 1/n !. To show the equivalence 
of the two definitions 11-13 and 11-14, we differentiate S n times 
with respect to s and then let s tend to zero, using first one and 
then the other expression for S; the terms with v < n vanish 
on differentiation, and those with v > n vanish for s — ■> 0, leaving 
only the term with v = n: 

\ r /»->-0 

and 

Comparing these two equations, we see that we obtain Equation 
11-13, so that the two definitions of H n (t) are equivalent. 
Equation 11-13 is useful for obtaining the individual functions, 
while Equation 11-14 is frequently convenient for deriving their 
properties, such as in the case we shall now discuss. 

To show that the functions we have defined above are the 
same as those used in the solution of the harmonic oscillator 
problem, we look for the differential equation satisfied by 
fln(£). It is first convenient to derive certain relations between 
successive Hermite polynomials and their derivatives. We 
note that since S = e p-i»-v* 9 its partial derivative with respect 
to s is given by the equation 

g - -*. - «* 

Similarly differentiating the series S = ^.—^ps", and equating 
the two different expressions for dS/ds f we obtain the equation 



»■(«).,-. 



_ -2< 8 - »^e. 



^J(n - 1)!° " y ° "^ n! 

n n 

or, collecting terms corresponding to the same power of s, 



m-llc] HARMONIC OSCILLATOR IN WAVE MECHANICS 79 
--^J ( n! (n — 1)1 n! J 

n 

Since this equation is true for all values of s, the coefficients of 
individual powers of s must vanish, giving as the recursion 
formula for the Hermite polynomials the expression 

JW0 - 2(H n (S) + 2n/7 n _ 1 (£) - 0. (11-15) 

Similarly, by differentiation with respect to {, we derive the 
equation 

|f - 2sS, 
which gives, in just the same manner as above, the equation 

n 

or 

».'(*) = ^P = 2ntf n __ 1 (£), (11-16) 

involving the first derivatives of the Hermite polynomials. 
This can be further differentiated with respect to £ to obtain 
expressions involving higher derivatives. 

Equations 11-15 and 11-16 lead to the differential equation for 
H n (0, for from 11-16 we obtain 

HM) = 2nHU(Q = 4n(n - l)ff-i(0, (11-17) 

while Equation 11-15 may be rewritten as 

# n (£) - 2£# n _ 1 ({) + 2(n - l)ff n - 2 (£) = 0, (11-18) 

which becomes, with the use of Equations 11-16 and 11-17, 

or 

Hi'(«) - 2«Hi(© + 2nH n (S) = 0. (11-19) 

This is just the equation, 11-6, which we obtained from the har- 
monic oscillator problem, if we put 2n in place of 1, as 

required by Equation 11-8. Since for each integral value of n 
this equation has only one solution with the proper behavior at 



80 THE SCHRODINQER WAVE EQUATION [m-llc 

infinity, the polynomials H n (g) introduced in Section 11a are 
the Hermite polynomials. 
The functions 

M*) = Nn-e 2 • H n (& 9 $ = Vi, (11-20) 

are called the Hermite orthogonal functions; they are, as we 
have seen, the wave functions for the harmonic oscillator. The 

ypl{x)dx = 1, i.e., which normalizes 

The functions are mutually orthogonal if the integral over 
configuration space of the product of any two of them vanishes: 

f*yn(x)+m(x)dx = 0, n*m. (11-22) 

To prove the orthogonality of the functions and to evaluate the 
normalization constant given in Equation 11-21, it is convenient 
to consider two generating functions: 



and 



s & s) = 2^r 8 " = ei ""~ v ' 



m, t) = 2%r' m = eP ~ lt ~ v> 



Using these, we obtain the relations 

n m 

= f g-0-iH-toHVt-Vdt = e 2 " f er«— »•<*({ - s-t) 

Considering coefficients of 8 n t m in the two equal series expansions, 

HniOHmifie-^dZ vanishes for m^n y and has the 

value 2 n n!\/ir for m = n, in consequence of which the functions 



m-llc] HARMONIC OSCILLATOR IN WAVE MECHANICS 81 

are orthogonal and the normalization constant has the value 
given above. 

The first few Hermite polynomials are 

Ho(0 =1 
ffitt) = 4? -2 

h*(q = 8? - m 

H A (Q = 16€ 4 - 48? + 12 

H b (0 = 32? - 160? + 120f (11-23) 

ff s (Q = 64? - 480? + 720? - 120 
JJ 7 (€) = 128? - 1344? + 3360? - 1680? 
H n (Q = 256? - 3584? + 13440? - 13440? + 1680 
H 9 (& = 512? - 9216? + 48384? - 80640? + 30240? 
Hio(& = 1024?° - 23040? + 161280? - 403200? + 302400? 

- 30240. 

The list may easily be extended by the use of the recursion 
formula, Equation 11-15. Figure 11-3 shows curves for the 
first few wave functions, i.e., the functions given by Equation 
11-20. 

By using the generating functions £ and T we can evaluate 
certain integrals involving \[/ n which are of importance. For 
example, we may study the integral which, as we shall later 
show (Sec. 40c), determines the probability of transition from the 
state n to the state m. This is 

x nm = tnKxdx = ^^ H n H m e-*m. (11-24) 

J— w OL J— ao 

Using S and T we obtain the relation 

n m 

/»+ 00 /»+ 00 

(* - s - t)d(S -8-t) 
e- ( «-'-°y(? -s-t). 

The first integral vanishes, and the second gives Vx- On 
expanding the exponential, we obtain 



82 THE SCHRODINOER WAVE EQUATION [Hi-lie 



V^( 






} 



Hence, comparing coefficients of s n t m , we see that x nm is zero 
except for m = n ± 1, its values then being 



Cn.« + i = ^^T (ll-25a) 



and 



It will be shown later that this result requires that transitions 
occur only between adjacent energy levels of the harmonic 
oscillator, in agreement with the conclusion drawn from the 
correspondence principle in Section 5c. 

Problem 11-1. Show that if V( — x) = V(x), with V real, the solutions 
yp n {x) of the amplitude equation {M* have the property that ^»( —x) — ± y// n (x) . 
Problem 11-2. Evaluate the integrals 

(X*)nm = J>„^m£ 2 dx, (x*) nn = ^n^mX i dx 1 (x*)nm = fMrnZ^dx, 

where ^ ft is a solution of the wave equation for the harmonic oscillator. 

Problem 11-3. Calculate the average values of x, x 2 , x*, and x* for a 
harmonic oscillator in the nth stationary state. Is it true that £* = (£)* 
or that P = (P) 2 ? What conclusions can be drawn from these results 
concerning the results of a measurement of x? 

Problem 11-4. Calculate the average values of p x and p* for a harmonic 
oscillator in the nth stationary state and compare with the classical values 
for the same total energy. From the results of this and of the last problem, 
compute the average value of the energy W = T -f- V for the nth 
stationary state. 

Problem 11-5. a. Calculate the zerp-point energy of a system consisting 
of a mass of 1 g. connected to a fixed point by a spring which is stretched 
1 cm. by a force of 10,000 dynes. The particle is constrained to move only 
in the x direction. 

b. Calculate the quantum number of the system when its energy is about 
equal to kT, where k is- Boltzmann's constant and T = 298° A. This corre- 
sponds to thermodynamic equilibrium at room temperature (Sec. 49). 

General References on Quantum Mechanics 

A. SoMMBRTBfcD: "Wave Mechanics," E. P. Dutton <fe Co., Inc., New 
York, 1930. 

E. XJ. Condon and P. M. Morse: "Quantum Mechanics," McGraw-Hill 
Book Company, Inc., New York, 1929. 



III-llc] HARMONIC OSCILLATOR IN WAVE MECHANICS 83 

A. E. Ruark and H. C. Urey: "Atoms, Molecules and Quanta," McGraw- 
Hill Book Company, Inc., New York, 1930. 

N. F. Mott: "An Outline of Wave Mechanics," Cambridge University 
Press, Cambridge, 1930. 

J. Frenkbl: "Wave Mechanics," Oxford University Press, 1933-1934. 

K. K. D arrow: Elementary Notions of Quantum Mechanics, Rev. Mod. 
Phys. 6, 23 (1934). 

E. C. Kemblb: General Principles of Quantum Mechanics, Part I, Rev. 
Mod. Phys. 1, 157 (1929). 

E. C. Kemble: "Fundamental Principles of Quantum Mechanics," Mc- 
Graw-Hill Book Company, Inc., 1937. 

E. C. Kemble and E. L. Hill: General Principles of Quantum Mechanics, 
Part II, Rev. Mod. Phys. 2, 1 (1930). 

S. Dushman: "Elements of Quantum Mechanics," John Wiley & Sons, 
Inc., J 938. 



CHAPTER IV 

THE WAVE EQUATION FOR A SYSTEM OF POINT 
PARTICLES IN THREE DIMENSIONS 

12. THE WAVE EQUATION FOR A SYSTEM OF POINT PARTICLES 

The Schrodinger equation for a system of N interacting point 
particles in three-dimensional space is closely similar to that for 
the simple one-dimensional system treated in the preceding 
chapter. The time equation is a partial differential equation 
in 3iV + 1 independent variables (the 3N Cartesian coordinates, 
say, of the N particles, and the time) instead of only two inde- 
pendent variables, and the wave function is a function of these 
3N + 1 variables. The same substitution as that used for the 
simpler system leads to the separation of the time equation into 
an equation involving the time alone and an amplitude equation 
involving the ZN coordinates. The equation involving the time 
alone is found to be the same as for the simpler system, so that 
the time dependency of the wave functions for the stationary 
states of a general system of point particles is the same as for 
the one-dimensional system. The amplitude equation, however, 
instead of being a total differential equation in one independent 
variable, is a partial differential equation in 3N independent 
variables, the 3A r coordinates. It is convenient to say that this 
is an equation in a 3iV-dimensional configuration space, meaning 
by this that solutions are to be found for all values of the 3N 
Cartesian coordinates X\ • • • z N from — oo to + °° • The 
amplitude function, dependent on these ZN coordinates, is said 
to be a function in configuration space. A point in configuration 
space corresponds to a definite value of each of the 3N coordi- 
nates Xi - • • z N , and hence to definite positions of the N particles 
in ordinary space, that is, to a definite configuration of the 
system. 

The wave equation, the auxiliary conditions imposed on the 
wave functions, and the physical interpretation of the wave 
functions for the general system are closely similar to those for 

84 



IV-12a] WAVE EQUATION FOR A SYSTEM OF PARTICLES 85 

the one-dimensional system, the only changes being those conse- 
quent to the increase in the number of dimensions of configura- 
tion space. A detailed account of the postulates made regarding 
the wave equation and its solutions for a general system of point 
^articles is given in the following sections, together with a dis- 
cussion of various simple systems for illustration. 

12a. The Wave Equation Including the Time. — Let us con- 
sider a system consisting of N point particles of masses mi, 
rri2i • • • , ™>n moving in three-dimensional space under the 
influence of forces expressed by the potential function V(xi 
2/i • • • Zn, t), xi • • • z N being the 3N Cartesian coordinates 
of the N particles. The potential function V, representing 
the interaction of the particles with one another or with an 
external field or both, may be a function of the 3iV coordinates 
alone or may depend on the time also. The former case, with 
V — V(xi - • • z N ), corresponds to a conservative system. 
Our main interest lies in systems of this type, and we shall soon 
restrict our discussion to them. 

We assume with Schrodinger that the wave equation for this 
system is 

N 

/i 2 %n l/d 2 * , d 2 * , a 2 *\ , TrT h d* /10 ,, 

-SH^im^ + M + *T ) + F * = "53 Hi' (12 " 1} 



This equation is often written as 

N 






It' 



in which v, ? is the Laplace operator or Laplacian for the ith 
particle. 1 In Cartesian coordinates, it is given by the expression 

Vl ~ dxf ^ dy? ^ dz\ 

The wave function ^ = V(xi • • • z*, t) is a function of the 
3JV coordinates of the system and the time. 

It will be noted that the Schrodinger time equation for this 
general system is formally related to the classical energy equation 
in the same way as for the one-dimensional system of the preced- 

^he symbol A is sometimes used in place of V 2 . The symbol V 2 is 
commonly read as del squared. 



86 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-12b 

ing chapter. The energy equation for a Newtonian system of 
point' particles is 

H(Vx l • ' ' Pz N , xi • • • z N) t) = T(p Xi • • • p, N ) + 

V(xi • • • z N) t) = TF, (12-2) 
which on explicit introduction of the momenta p xi . . . p*^ 
becomes 

ff(p. t • • • p v * • • ■ ** = 2 2 i ( ^ + ^ + ^ + 

7(si ■ • • zn, t) = F. (12-3) 
We now arbitrarily replace the momenta p^ • • • p ZN by 

the differential operators -—.—-... -_ - — , respectively, 

2in dxi 2-ki dz N 

h rk 

and W by the operator — jr— . — , and introduce the function 

2m dt 

V(xi • • • z N , t) on which these operators can operate. The 

equation then becomes 

„/ h d h d \ 

H \2ritel ' ' ' 2ridT N ,Xl ' ' '"''J* 

= __^^! V 2* + w = _ A «* (12-4) 

which is identical with Equation 12-1. Just as for the one- 
dimensional case, the wave equation is often symbolically 
written 

HV = IF*. (12-5) 

The discussion in Section 9a of the significance of this formal 
relation is also appropriate to this more general case. 

12b. The Amplitude Equation. — Let us now restrict our atten- 
tion to conservative systems, for which V is a function of the 3iV 
coordinates only. To solve the wave equation for this case, 
we proceed exactly as in the simpler problem of Section 96, 
investigating the solutions * of the wave equation which can be 
expressed as the product of two functions, one of which involves 
only the time and the other only the 3N coordinates: 

*(xi •••**,*)= *(xi • • • zs)<p(t). (12-6) 

On introducing this expression in Equation 12-1, the wave equa- 



IV-12b] WAVE EQUATION FOR A SYSTEM OF PARTICLES 87 

tion can be separated into two equations, one for <p(t) and one 
for }p(xi • • • Zn). These equations are 

at n 

"^ K ) (12-7) 



-£2^ + v + = F ^ 



i-1 

The second of these is often written in the form 

N 



2^ + 8 -£ {W - m = °- (1 *~ 8) 



*-i 



This is Schrodinger's amplitude equation for a conservative 
system of point particles. 

The auxiliary conditions which must be satisfied by a solution 
of the amplitude equation in order that it be an acceptable wave 
function are given in Section 9c. These conditions must hold 
throughout configuration space, that is, for all values between 
— oo and + oo for each of the ZN Cartesian coordinates of the 
system. Just as for the one-dimensional case, it is found that 
acceptable solutions exist only for certain values of the energy 
parameter W. These values may form a discrete set, a con- 
tinuous set, or both. 

It is usually found convenient to represent the various succes- 
sive values of the energy parameter and the corresponding ampli- 
tude functions by the use of SN integers, which represent 3N 
quantum numbers rti • • • n ZN , associated with the 3N coordi- 
nates. The way in which this association occurs will be made 
clear in the detailed discussion of examples in the following 
sections of this chapter and in later chapters. For the present 
let us represent all of the quantum numbers n\ • • ■ n ZN by 
the one letter n, and write instead of W ni • * • % w an< i iki " * ' »w 
the simpler symbols W n and \fr n . x^ K > 

The equation for <p(t) gives on integration L ^ 

.Wn 



-2wiZ-rt 



_„.£< _ V J (12-9) 
<p(t) = e * , v 

exactly as for the one-dimensional system. The various particu- 
lar solutions of the wave equation are hence 



88 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-12d 

*n(xi • • • z N , t) = + n (xi ' • • z N )e h . (12-10) 

These represent the various stationary states of the system. The 
general solution of the wave equation is 

¥(xi • • • Zn> t) = ^a n ^ n (xi • • • z N , t) = 

n 

-2lrt-r-f 

2/a»(aa • • • z^)e , (12-11) 

n 

in which the quantities a n are constants. The symbol V repre- 

n 

sents summation for all discrete values of W n and integration over 
all continuous ranges of values. 

12c. The Complex Conjugate Wave Function W*(xi • • • zn> 
0- — The complex conjugate wave function V*(xi • • • zx, t) 
is a solution of the conjugate wave equation 

N 



-£2£***<* -•*«*> + 



t-1 

ssl**** •••«*•'>■ (12 " 12) 

The general solution of this equation for a conservative system is 

**(Zi • • • Zn, t) = ^*^*(Xi • • • Sat, = 

n 



2a.v:(*. ■ • • m« * • (12 " 13) 



12d. The Physical Interpretation of the Wave Functions. — 

The physical interpretation of the wave functions for this general 
system is closely analogous to that for the one-dimensional system 
discussed in Section 10. We first make the following postulate, 
generalizing that of Section 10a: 

The quantity ^*(xi • • • z N , t)^(xi • • • z N , t)dx\ • • • dz N is 
the probability that the system in the physical situation represented 
by the wave function ty(x i • • • Zn, t) have at the timet the configura- 
tion represented by a point in the volume element dx\ • • • dzN of 
configuration space. ¥ *¥ thus serves as a probability distribution 
function for the configuration of the system. 



IV-12d] WAVE EQUATION FOR A SYSTEM OF PARTICLES 89 

The function V(xi • • • z N , t) must then be normalized to 
unity, satisfying the equation 

S**(zi • • • z Ny 0*(*i • • z N , t)dr = 1, (12-14) 

in which the symbol dr is used to represent the volume element 
dxi • • • dz N in configuration space, and the integral is to* be 
taken over the whole of configuration space. (In the remaining 
sections of this book the simple integral sign followed by dr 
is to be considered as indicating an integral over the whole of 
configuration space.) It is also convenient to normalize the 
amplitude functions \p n (xi • • • z N ), according to the equation 

JVn*(*i • • • z»)M*i • • • *»)d* = 1. (12-15) 

It is found, as shown in Appendix III, that the independent 
solutions of any amplitude equation (just as for the one-dimen- 
sional case) can be chosen in such a way- that any two of them are 
orthogonal, satisfying the orthogonality equation 

jVtffri • • • z N )+ n (*i ' ' • z N )dr = 0, m^n. (12-16) 
A wave function ty(xi • • • z N > t) = ^\a n ^ n {xi • • • zs, t) is 

n 

then normalized if the coefficients a n satisfy the equation 

2<*X = 1. (12-17) 

n 

An argument analogous to that of Section 10b shows that the 

wave functions V n (xi • • • z N , t) = \p n (xi • • • z N )e h give 
probability distribution functions which are independent of the 
time and hence correspond to stationary states. 

A more general physical interpretation can be given the wave 
functions, along the lines indicated in Section 10c, by making 
the postulate that the average value of the dynamical function 
G(p Xl • • • p tJf) xi • • • z N , t) predicted for a system in the 
physical situation represented by the wave function V(xi • • • z N , 
t) is given by the integral 

in which the operator G, obtained from G(p Xl 



h 
2iri 


d 

dz N y 


Xi • • 


• z Ny t) 


• • 


' **: 


t 0*> 


(12-18) 


[P*x 


. . . 


V*N> Zl 


• • • Zjfj 



90 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-13 

h A h A 

t) by replacing p„ . . . p,„by_— . . . ——respectively, 

operates on the function ^(xi • • • z Ny t) and the integration is 
extended throughout the configuration space of the system. 
Further discussion of the physical interpretation of the wave 
functions will be found in Chapter XV. 

13. THE FREE PARTICLE 

A particle of mass m moving in a field-free space provides the 
simplest application of the Schrodinger equation in three dimen- 
sions. Since V is constant (we choose the value zero for con- 
venience), the amplitude equation 12-8 assumes the following 
form: 

W + ^Wi = 0, (13-1) 

or, in Cartesian coordinates, 

a* + W i+ & + "»~ W * = ( } 

This is a partial differential equation in three independent 
variables x, y, and z. In order to solve such an equation it is 
usually necessary to obtain three total differential equations, 
one in each of the three variables, using the method of separation 
of variables which we have already employed to solve the 
Schrodinger time equation (Sec. 96). We first investigate the 
possibility that a solution may be written in the form 

Mx, y, z) = X(x) • Y(y) • Z(z), (13-3) 

where X{x) is a function of x alone, Y(y) a function of y alone, 
and Z(z) a function of z alone. If we substitute this expression 
in Equation 13-2, we obtain, after dividing through by i/s the 
equation 

ld " X + iS + iS+^-0. (.3-4) 



X dx 2 ^ Y dy* ' Z dz* ' h 2 

Since X is a function only of x, the first term does not change 
its value when y and z change. Likewise the second term is 
independent of x and z and the third term of x and y. Never- 
theless, the sum of these three terms must be equal to the con- 

stant jt"W f° r an y c h° ice of x, y, z. By holding y and z 



IV-13] THE FREE PARTICLE 91 

fixed and varying x, only the first term can vary, since the others 

do not depend upon x. However, since the sum of all the 

terms is equal to a constant, we are led to the conclusion that 

1 d 2 X 

v ~J~2 * s independent of x as well as of y and z, and is therefore 

itself equal to a constant. Applying an identical argument to 
the other terms, we obtain the three ordinary differential 
equations 

I^-fc 1*1 -k and !*?-* (13-5) 

X dx* ~ Kx ' Y dy* Kv ' a Q Z dz 1 ~ *" K16 d) 

with the condition 

k x + h, + k, = -^jrW. (13-6) 

It is convenient to put k x = r^-J^x, which gives the equation 



in z the form 



d? + 1^ Z = °- (13_7) 



This is now a total differential equation, which can be solved 
by familiar methods. As may be verified by insertion in the 
equation, a solution is 

X(x) = N x sin <^V2mW x (x - xo)Y (13-8) 

Since it contains two independent arbitrary constants N x and x 0j 
it is the general solution. It is seen that the constant x defines 
the location of the zeros of the sine function. The equations for 
Y and Z are exactly analogous to Equation 13-7, and have the 
solutions 



Y(y) = N y sin ^V2mW w (y - y<>)}> J 
Z(z) = N 8 sin <jV2mW z (z - z ) }• 



(13-9) 



The fact that we have been able to obtain the functions 
X, Y, and Z justifies the assumption inherent in Equation 13-3. 
It can also be proved 1 that no other solutions satisfying the 

1 The necessary theorems are given in R. Courant and D. Hilbert, 
"Methoden der mathematischen Physik," 2d ed., Julius Springer, Berlin, 
1931. 



92 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-13 

boundary conditions can be found which are linearly independent 
of these, i.e., which cannot be expressed as a linear combination 
of these solutions. 

The function \p must now be examined to see for what values of 
W = W x + W y + W, it satisfies the conditions for an acceptable 
wave function given in Section 9c. Since i^he sine function is 
continuous, single-valued, and finite for all real values of its 
argument, the only restriction that is placed on W is that W x , 
W y , W» and therefore W be positive. We have thus reached the 
conclusion that the free particle has a continuous spectrum of 
allowed energy values, as might have been anticipated from the 
argument of Section 9c. 

The complete expression for the wave function corresponding 
to the energy value 

W = W x + W v + W, (13-10) 

is 

Wz> V>z) = N sin \-j-\/ZrriW x {x — x )f 
• sin ft V^mWviy - »o)} • sin <^V2mWz(z - zt>)\, (13-11) 

in which N is a normalization constant. The problem of the 
normalization of wave functions of this type, the value of which 
remains appreciable over an infinite volume of configuration 
space (corresponding to a continuous spectrum of energy values), 
is a complicated one. Inasmuch as we shall concentrate our 
attention on problems of atomic and molecular structure, with 
little mention of collision problems and other problems involving 
free particles, we shall not discuss the question further, contenting 
ourselves with reference to treatments in other books. 1 

In discussing the physical interpretation of the wave functions 
for this system, let us first consider that the physical situation is 
represented by a wave function as given in Equation 13-11 
with W y and W z equal to zero and W x equal to W. The func- 

1 A. Sommerpeld, "Wave Mechanics," English translation by H. L. 
Brose, pp. 293-295, E. P. Dutton & Co., Inc., New York, 1929; Rxjark and 
Urey, "Atoms, Molecules, and Quanta/' p. 541, McGraw-Hill Book Com- 
pany, Inc., New York, 1930. 



IV-13] THE FREE PARTICLE 93 

tion 1 <&{x, y, z, t) = N sin \-j--\/2mW(x — Xo)/-6 h is then a 

set of standing waves with wave fronts normal to the x axis. 
The wave length is seen to be given by the equation 

x - v^f ' (13 " 12) 

In classical mechanics the speed v of a free particle of mass m 
moving with total energy W is given by the equation Y^mv 1 = W. 
A further discussion of this system shows that a similar inter- 
pretation of W holds in the quantum mechanics. Introducing 
v in place of W in Equation 13-12, we obtain 

X = — • (13-13) 

mv 

This is the de Broglie expression 2 for the wave length associated 
with a particle of mass m moving with speed v. 

It is the sinusoidal nature of the wave functions for the free 
particle and the similar nature of the wave functions for other 
systems which has caused the name wave mechanics to be applied 
to the theory of mechanics which forms the subject of this book. 
This sinusoidal character of wave functions gives rise to experi- 
mental phenomena which are closely similar to those associated 
in macroscopic fields with wave motions. Because of such 
experiments, many writers have considered the wavelike char- 
acter of the electron to be more fundamental than its corpuscular 
character, but we prefer to regard the electron as a particle and 
to consider the wavelike properties as manifestations of the 
sinusoidal nature of the associated wave functions. Neither 
view is without logical difficulties, inasmuch as waves and 
particles are macroscopic concepts which are difficult to apply to 
microscopic phenomena. We shall, however, in discussing the 
results of wave-mechanical calculations, adhere to the particle 
concept throughout, since we believe it is the simplest upon 
which to base an intuitive feeling for the mathematical results 
of wave mechanics. 

1 It can be shown that the factors involving y and x in Equation 13-11 
approach a constant value in this limiting case. 
1 L. db Broglib, Thesis, 1924; Ann. de phys. 3, 22 (1925). 



94 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-13 

The wave function which we have been discussing corresponds 
to a particle moving along the x axis, inasmuch as a calculation 

of the kinetic energy T x = k~pI associated with this motion 

shows that the total energy of the system is kinetic energy of 
motion in the x direction. This calculation is made by the 
general method of Section 12d. The average value of T x is 



- &f*'{M&* d ' 



= w x , 

or, since in this case we have assumed W x to equal W, 

T x = W. 

Similarly we find T r x = W r = (T x ) r , which shows, in accordance 
with the discussion of Section 10c, that the kinetic energy of 
motion along the x axis has the definite value W, its probability 
distribution function vanishing except for this value. 

On the other hand, the average value of p x itself is found on 
calculation to be zero. The wave function 

N sin \-j-\/2mW(x — x )fe ** h 

hence cannot be interpreted as representing a particle in motion 
in either the positive or negative direction along the x axis but 
rather a particle in motion along the x axis in either direction, 
the two directions of motion having equal probability. 

The wave function N cos \-j-\/2mW(x — xo)fe r%h differs 

from the sine function only in the phase, the energy being the 
same. The sum and difference of this function and the sine func- 
tion with coefficient i are the complex functions 

2Tt r= — si/ n 2riW. 2W /s — =.. . 2-iriW \ 

^7^"-^-^ and N > e ~^ m ^-*\-— \ 

which are also solutions of the wave equation equivalent to the 
sine and cosine functions. These complex wave functions 
represent physical situations of the system in which the particle 
is moving along the x axis in the positive direction with the 



IV-14] THE PARTICLE IN A BOX 95 

definite momentum p x =» \/2mW or p x = — s/2mW, the 
motion in the positive direction corresponding to the first of 
the complex wave functions and in the negative direction to the 
second. This is easily verified by calculation of pi and pj for 
these wave functions. 

The more general wave function of Equation 13-11 also 
represents a set of standing plane waves with wave length 
X = h/\/2mW, the line normal to the wave fronts having the 

direction cosines \/W x /W y \ZW y /W y and \ZW Z /W relative to the 
x, y, and z axes. 

Problem 13-1. Verify the statements of the next to the last paragraph 
regarding the value of p x . 

14. THE PARTICLE IN A BOX 1 

Let us now consider a particle constrained to stay inside of a 
rectangular box, with edges a, b> and c in length. We can repre- 
sent this system by saying that the potential function V(Xj y, z) 
has the constant value zero within the region < x < a, 
< y < by and < z < c y and that it increases suddenly in 
value at the boundaries of this region, remaining infinitely 
large everywhere outside of the boundaries. It will be found 
that for this system the stationary states no longer correspond to 
a continuous range of allowed energy values, but instead to a 
discrete set, the values depending on the size and shape of the 
box. 

Let us represent a potential function of the type described as 

V(x, y, z) = V x (x) + V v (y) + V,{z), (14-1) 

the function V x (x) being equal to zero for < x < a and to 
infinity for x < or x > a, and the functions V v (y) and V z {z) 
showing a similar behavior. The wave equation 

dV aV dV 8ir 2 m 
dx 2 + dy 2 + dz 2 + h 2 

{W - V x (x) - V y (y) - V&)}* = (14-2) 

is separated by the same substitution 

+(x, y } z) = X(x) • Y(y) • Z(z) (14-3) 

1 Treated in Section Qd by the methods of the old quantum theory. 



96 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-14 



as for the free particle, giving three total differential equations, 
that in x being 



d*X , &r 2 m 



dx* 



+ 



{W x - V x (x)}X = 0. 



(14-4) 



In the region < x < a the general solution of the wave 
equation is a sine function of arbitrary amplitude, frequency, and 
phase, as for the free particle. Several such functions are repre- 
sented in Figure 14-1. All of these are not acceptable wave 



w, 



fr 



o-- 



w y 



V x (x) 




Fig. 14-1. — The potential-energy function V x (x) and the behavior of X(x) near 

the point x — a. 

functions, however; instead only those sine functions whose 
value falls to zero at the two points x = and x — a behave 
properly at the boundaries. To show this, let us consider the 
behavior of Curve A as x approaches and passes the value a, 
using the type of argument of Section 9c. Curve A has a finite 
positive value as x approaches a, and a finite slope. Its curvature 
is given by the equation 

d*X 
dx 2 



8* 2 m 



A 2 



\W X - V x {x))X. 



(14-5) 



At the point x = a the value of V(x) increases very rapidly and 
without limit, so that, no matter how large a value the constant 
W, has, Wx — V, becomes negative and of unbounded magni- 



IV-141 



THE PARTICLE IN A BOX 



97 



tude. The curvature or rate of change of the slope consequently 
becomes extremely great, and the curve turns sharply upward 
and experiences the infinity catastrophe. This can be avoided 
in only one way; the function X(x) itself must have the value 
zero at the point x = a, in order that it may then remain bounded 
(and, in fact, have the value zero) for all larger values of x. 

Similarly the sine function must fall to zero at x = 0, as shown 
by Curve C. An acceptable wave function X(x) is hence a 
sine function with a zero at x = and another zero at x = a, 




Fig. 14-2. — The wave functions Xn x (x) and probability distribution functions 
[Xn x (x)]* for the particle in a box. 

thus having an integral number of loops in this region. The 
phase and frequency (or wave length) are consequently fixed, 
and the amplitude is determined by normalizing the wave func- 
tion to unity. Introducing the quantum number n x as the 
number of loops in the region between and a, the wave length 
becomes 2a/n x , and the normalized X(x) function is given by 
the expression 



X Vx (x) = 
with 



2 . n x TX 

- sin ; 

a a 



n x = 1, 2, 3, 



nnm 8ma 2 



< x < a, (14-6) 



(14-7) 



98 WAVE EQUATION FOR A SYSTEM OF PARTICLES [VI-H 

The first four wave functions Xi(x), • • • , X 6 (x) are represented 
in Figure 14-2, together with the corresponding probability 
distribution functions { X n *(x) } 2 . 

A similar treatment of the y and z equations leads to similar 
expressions for Y nv (y) and Z n ,(z) and for W y and W t . The com- 





4 


/ 












































2H 






x 


116 


















2; 






2 




£ 


/ 
































"" 






2q 






2 








IIS 






^ 






2 






2 






^ 


£ 


/ 
































^ 






2 












/ 


114 






^ 






z 












^ 




^ 


































2 






2q 






2: 






/ 


113 






^ 






z 






2 






z 




£ 


/ 
































^ 






^d 














w? 






2 






2 












^ 




121 


l.M 


























n ~ M hi 








/ 








/ 






n x 


„ y ,, 2 * M, 


^ 


^ 


s 


s 






^ 




s 


* 211 ^jli_- —"411 511 
% - l 4ll 

r 








«~ """" 







0. . 'a 
riojin 

Fig. 14-3. — A geometrical representation of the energy levels for a particle in a 

rectangular box. 

plete wave function \p nx n v nXx, V, *) has the form, for values of x, 
y, and z inside the box, 

^nxn.n,^, 2/, 2) = A -^ sin sm —^ sin > (14-8) 

\aoc .a b c 

withn* = 1,2,3, • • • ;n y = 1,2,3, ■ • • ;n, = 1,2,3, • • • ; 
and 

WW ■= JF n , + TF„„ + W n , = ^ + Jf + £). (14-9) 

The wave function ^ nxByn , can be described as consisting of 
standing waves along the x, y, and z directions, with n x + 1 
equally spaced nodal planes perpendicular to the x axis (begin- 
ning with x = and ending with x — a), n„ 4- 1 nodal Dlanes 



IV-14] 



THE PARTICLE IN A BOX 



99 



perpendicular to the y axis, and n t + 1 nodal planes perpendicular 
to the z axis. 

The various stationary states with their energy values may be 
conveniently represented by means of a geometrical analogy. 
Using a system of Cartesian coordinates, let us consider the 



27 
26 
24 

22 
21 

19 
18 
17 



8ma 



c 2 WJ 



n x n y n z 

•p=4 51le+c,333 

•P =6 431 etc. 

•p =3 422 e+c. 

•P =3 332 e+c. 

•P =6 421 e+c. 

■P =3 33!e+c. 

•P =3 4lle+c. 

•P =3 322 e+c. 



•P =6 321,132,213,312,231,123 

• P = I 222 

P =3 311,131,113 

•P =3 221,122,212 



• P =3 211.121,112 



p = i Hi 



Fia. 14-4. 



-Energy levels, degrees of degeneracy, and quantum numbers for a 
particle in a cubic box. 



lattice whose points have the coordinates n x /a y n v /b, and n t /c, 
with n x = 1, 2, • • • ; n y = 1, 2, • • • ; and n ? = 1, 2 • • • . 
This is the lattice defined in one octant about the origin by the 
translations 1/a, 1/6, and 1/c, respectively; it divides the octant 
into unit cells of volume 1/abc (Fig. 14-3). Each point of the 
lattice represents a wave function. The corresponding energy 
value is~ 



h 2 
W = — 7 2 



(14-10) 



100 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-15 

in which l nx n>n. is the distance from the origin to the lattice point 
n x nyn t , given by the equation 



w. = vP+f+f ■ ( 14 - n ) 

In case that no two of the edges of the box a, 6, and c are in 
the ratio of integers, the energy levels corresponding to various 
sets of values of the three quantum numbers are all different, 
with one and only one wave function associated with each. 
Energy levels of this type are said to be non-degenerate. If, 
however, there exists an integral relation among a, 6, and c, 
there will occur certain values of the energy corresponding to 
two or more distinct sets of values of the three quantum numbers 
and to two or more independent wave functions. Such an energy 
level is said to be degenerate, and the corresponding state of the 
system is called a degenerate state. For example, if the box is a 
cube, with a = b = c, most of the energy levels will be degener- 
ate. The lowest level, with quantum numbers 111 (for n Xf 
Uyj n z , respectively) is non-degenerate, with energy 3/i 2 /8ma 2 . 
The next level, with quantum numbers 211, 121, and 112 and 
energy 6/i 2 /8ma 2 , is triply degenerate. Successive levels, with 
sets of quantum numbers and degrees of degeneracy (represented 
by p), are shown in Figure 14-4. The degree of degeneracy 
(the number of independent wave functions associated with a 
given energy level) is often called the quantum weight of the 
level. 

15. THE THREE-DIMENSIONAL HARMONIC OSCILLATOR IN 
CARTESIAN COORDINATES 

Another three-dimensional problem which is soluble in Car- 
tesian coordinates is the three-dipiensional harmonic oscillator, 
a special case of which, the isotropic oscillator, we have treated 
in Section la by the use of classical mechanics. The more general 
system consists of a particle bound to the origin by a force whose 
components along the x, y, and z axes are equal to — k x x, —Jc y y, 
and — k t z, respectively, where k x> k y , k z are the force constants 
in the three directions and x, y, z are the components of the 
displacement along the three axes. The potential energy 
is thus 

V = HktZ* + y 2 k y y* + %k z z\ (15^1) 



IV-15] THREE-DIMENSIONAL HARMONIC OSCILLATOR 101 



which, on introducing instead of the constants k x> k v , k t their 
expressions in terms of the classical frequencies v x , v v , v» } becomes 



smce 



V = 27r 2 m(v£c 2 + v\y 2 + v 2 z 2 ), 



k x = 4ir 2 mvly 
k y = 47r 2 mv 2 



(16-2) 



(15-3) 



k, = 4ir 2 m*/ 2 . 



The general wave equation 12-8 thus assumes for this problem 
the form 



dx 2 



by 2 



av 

dz 2 



^ + ^ + ^ + S ^W- 2* 2 m(v 2 x> + v\y 2 + &*)) + 



h 2 



which, on introducing the abbreviations 

&w 2 m rTr 
W, 



and 



simplifies to the equation 







h 2 


a x 


= 


4w 2 m 

h ' 


Oy 


= 


47r 2 m 

h ' 


rt. 




4r 2 m 
1 



o, 

(16-4) 



(15-5a) 
(15-56) 
(16-5c) 

(15-5d) 



To solve this equation we proceed in exactly the same manner 
as in the case of the free particle (Sec. 13) ; namely, we attempt to 
separate variables by making the substitution 

*(*, y, z) = X(x) • 7(2/) • Z(z). (15-7) 

This gives, on substitution in Equation 15-6 and division of the 
result by ^, the equation 

(15-8) 



102 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-15 

It is evident that this equation has been separated into terms 
each of which depends upon one variable only; each term is 
therefore equal to a constant, by the argument used in Section 13. 
We obtain in this way three total differential equations similar 
to the following one : 

^) + (x, _ a l x *)X(x) = 0, (15-9) 

in which X* is a separation constant, such that 

X* + X* + Xz = X. (15-10) 

Equation 15-9 is the same as the wave equation 11-2 for the 
one-dimensional harmonic oscillator which was solved in Section 
11. Referring to that section, we find that X(x) is given by the 
expression 

X(x) = N n ,e 2 Hn (V&c) (15-11) 

and that X x is restricted by the relation 

X, = (2n x + 1)«„ (15-12) 

in which the quantum number n x can assume the values 0, 1, 
2, • • • . Exactly similar expressions hold for Y(y) and Z[z) 
and for X^ and X*. The total energy is thus given by the equation 

W nx n v n, = A { (n x + Y 2 ) V x + (lly + %) V y + (n z + %)».}, (15-13) 

and the complete wave function by the expression 

tn z n v nXx> V, «) = 

^.n.*/^-^"^ (15-14) 

The normalizing factor has the value 

_ J (ojw)" V* n*-i»« 

For the special case of the isotropic oscillator, in which 
v x = v y = v, = vo and a x = oty = a e , Equation 15-13 for the energy 
reduces to the form 

W = (n x + ny + n z + %)hv - (n + %)hv . (15-16) 

n = n x + riy + n z may be called the total quantum number. 
Since the energy for this system depends only on the sum of the 



IV-16] 



CURVILINEAR COORDINATES 



103 



quantum numbers, all the energy levels for the isotropic oscilla- 
tor, except the lowest one, are degenerate, with the quantum 

weight ~ lyLJL—L. Figure 15-1 shows the first few energy 



II 


n»4 


— Oslt 


1 






9 


n-3 


- p-10- 


300,030,003, 
210,011 JOZ, 


2 




201,120,012, 
1/1. ' 


W 7 


n = 2 


- d«6 $200,020,002, 


bv > 


Z 




p *\iio',ioi',on. 




5 


n»l 


wmm n»^ //9/7 /)//) /)/0/ 


2 






3 


n«0 




2 
















Fig. 15-1.- 



-Energy levels, degrees of degeneracy, and quantum numbers for the 
three-dimensional isotropic harmonic oscillator. 



levels, together with their quantum weights and quantum 
numbers. 



16. CURVILINEAR COORDINATES 

In Chapter I we found that curvilinear coordinates, such as 
spherical polar coordinates, are more suitable than Cartesian 
coordinates for the solution of many problems of classical 
mechanics. In the applications of wave mechanics, also, it is 
very frequently necessary to use different kinds of coordinates. 
In Sections 13 and 15 we have discussed two different systems, 
the free particle and the three-dimensional harmonic oscillator, 
whose wave equations are separable in Cartesian coordinates. 
Most problems cannot be treated in this manner, however, since 
it is usually found to be impossible to separate the equation into 
three parts, each of which is a function of one Cartesian coordi- 
nate only. In such cases there may exist other coordinate 
systems in terms of which the wave equation is separable, so 
that by first transforming the differential equation into the proper 



104 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-16 

coordinates the same technique of solution may often be 
applied. 

In order to make such a transformation, which may be repre- 
sented by the transformation equations 

x = f(u, v, w), (16-la) 

V = g(u, v, w), (16-16) 

z = h(u, v f w), (16-lc) 

it is necessary to know what form the Laplace operator V 2 assumes 
in the new system, since this operator has been defined only in 
Cartesian coordinates by the expression 

d 2 d 2 d 2 

*' - 1? + £■ + y (i6 - 2) 

The process of transforming these second partial derivatives is a 
straightforward application of the principles of the theory of 
partial derivatives and leads to the result that the operator V 2 
in the orthogonal coordinate system uvw has the form 



V2 = 1 j d (q v q w d\ 6 /q u q w d\ ±(qyqv _d\) 
quq v qw\du\ q u bu) dv\ q v dvj dw\ q w dw/j 



in which 



(16-3) 



Equation 16-3 is restricted to coordinates u, v, w which are 
orthogonal, that is, for which the coordinate surfaces represented 
by the equations u = constant, v = constant, and w = constant 
intersect at right angles. All the common systems are of this 
type. 

The volume element dr for a coordinate system of this type 
is also determined when q u , q v , and q w are known. It is given by 
the expression 

dr = dxdydz = q u q v q w dudvdw. (16-5) 

In Appendix IV, q u , q v , qw, and v 2 itself are given for a number of 
important coordinate systems. 



IV-17] THREE-DIMENSIONAL HARMONIC OSCILLATOR 105 

Mathematicians 1 have studied the conditions under which the 
wave equation is separable, obtaining the result that the three- 
dimensional wave equation can be separated only in a limited 
number of coordinate systems (listed in Appendix IV) and then 
only if the potential energy is of the form 

V = qu$u(u) + q v $v(v) + q w <$>w(w), 

in which 4> w (w) is a function of u alone, $ v (v) of v alone, and 
$ w (w) of w alone. 

17. THE THREE-DIMENSIONAL HARMONIC OSCILLATOR IN 
CYLINDRICAL COORDINATES 

The isotropic harmonic oscillator in space is soluble by separa- 
tion of variables in several coordinate systems, including Car- 



..** 



((>*%& 



-*Y 



Fia. 17-1. — Diagram showing cylindrical coordinates. 

tesian, cylindrical polar, and spherical polar coordinates. We 
shall use the cylindrical system in this section, comparing the 
results with those obtained in Section 15 with Cartesian 
coordinates. 

Cylindrical polar coordinates p, <p, z, which are shown in' 
Figure 17-1, are related to Cartesian coordinates by the equations 
of transformation 



x = p cos <p,\ 
y = psin ip} 
z = z. \ 



(17-1) 



1 H. P. Robertson, Mathematische Annalen 98, 749 (1928); L. P. Eisbn- 
hart, Ann. Mathematics 35, 284 (1034). 



106 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-17 

Reference to Appendix IV shows that V 2 in terms of p, <p, z has 
the form 

Vi.i i/p^ + I^+*!. (17-2) 

p dp\ p d P J ^ p 2 d<p 2 ^ dz* u ; 

Consequently, the wave equation 15-4 for the three-dimensional 
harmonic oscillator becomes 



p dp\dpj p 



dhp 6V 87r 2 m 

1 ^o I 



p 2 dv> 2 az 2 T h 2 

{W - Wm{y\p 2 + viz 2 )}* = 0, (17-3) 

when we make v x = v y = v (only in this case is the wave equa- 
tion separable in these coordinates). Making the substitutions 

x 87r 2 m T1T /ihT , , 

X = -rrW, (17-4a) 



and 



we obtain the equation 



a = —r-Po, (17-46) 



a* = — r-y., (17-4c) 



p ^VW + ? a7 2 + a* 2 + (x "" ap - °*^ = a (17 " 5) 

Pursuing the method used in Section 15, we try the substitution 

* = P( P ) • *(*) • Z(*), (17-6) 

in which P(p) depends only on p, <!>(<?) only on <p> and Z(z) only 
on z. Introduction of this into Equation 17-5 and division by \p 
leads to the expression 

1 d/dP\ 1 d 2 <i> ld 2 Z 22 

Fp^VW + ^d? + z^ + x ^ ap a ** =a (17 " 7) 

The terms of this equation may be divided into two classes: 
those which depend only on z and those which depend only on 
p and <p. As before, since the two parts of the equation are func- 
tions of different sets of variables and since their sum is constant, 



IV-17] THREE-DIMENSIONAL HARMONIC OSCILLATOR 107 

each of the two parts must be constant. Therefore, we obtain 
two equations 

Jf + (A, - a*,z*)Z = (17-8) 



dz 



and 



with 



rpdh%) + 7*d?- af > t + x = ' (17 ~ 9) 



X' + X, = X 

The first of these is the familiar one-dimensional harmonic 
oscillator equation whose solutions 

Z n ,(z) = N nz e~^H ni (V^zz) (17-10) 

are the Hermite orthogonal functions discussed in Section lie. 
As in the one-dimensional problem, the requirement that the 
wave function satisfy the conditions of Section 9c restricts the 
parameter X* to the values 

X* = (2n z + l)o., n z = 0, 1, 2, • • • . (17-11) 

Equation 17-9, the second part of the wave equation, is a 
function of p and v and so must be further separated. This 
may be accomplished by multiplying through by p 2 . The 
second term of the resulting equation is independent of p; it is 
therefore equal to a constant, which we shall call —m 2 . The 
two equations we obtain are the following: 

d 2 * 



and 



The first of these is a familiar equation whose normalized solution 

is 1 

<!>(<?) = — 7= e imip . (17-14) 

Inasmuch as e im * is equal to cos trap + i sin m<p, we see that for 
arbitrary values of the separation constant m this function is 

1 Instead of the exponential, the forms &(<p) = N cos m<p and N sin nap 
may be used. See Section 186, Chapter V. 



108 WAVE EQUATION FOR A SYSTEM OF PARTICLES (IV-17 

not single- valued; that is, <£ does not have the same value for 
if> = and for <p = 2r, which correspond to the same point in 
space. Only when m is a positive or negative integer or zero is 
<£ single-valued, as is required in order that it be an acceptable 
wave function (Sec. 9c) ; m must therefore be restricted to such 
values. <p is called a cyclic coordinate (or ignorable coordinate), 
these names being applied to a variable which does not occur 
anywhere in the wave equation (although derivatives with respect 
to it do appear). Such a coordinate always enters the wave 
function as an exponential factor of the type given in Equation 
17-14. * 

The equation for P(p) may be treated by the same general 
method as was employed for the equation of the linear harmonic 
oscillator in Section 11a. The first step is to obtain an asymp- 
totic solution for large values of p, in which region Equation 
17-13 becomes approximately 

~ - <* 2 p 2 P = 0. (17-15) 

The asymptotic solution of this is e 2 , since this function 
satisfies the equation 

which reduces to 17-15 for large values of p. Following the 
reasoning of Section 11a, we make the substitution 

P(p) = e~^/(p) (17-16) 

in Equation 17-13. From this we find that / must satisfy the 

equation 

1 m 2 

f" - 2a P f + if + (X' - 2a)/ - ^f = 0. (17-17) 

As before, it is convenient to replace p by the variable 

* - V^P (17-18) 

and/(p) by F(£), a process which gives the equation 

% -*5HS +£- 2 -f>= ' <»-*> 

1 Condon and Morse, "Quantum Mechanics," p. 72. 



IV-17] THREE-DIMENSIONAL HARMONIC OSCILLATOR 109 

We could expand F directly as a power series in £, as in Section 
11a. This is not very convenient, however, because the first 
few coefficients would turn out to be zero. Instead, we make 
the substitution 

00 

F(S) = ?X a '? = a °*' + ai *' +1 + ' ' ' ' (17 " 20) 

in which s is an undetermined parameter and a is not equal 
to zero. 

This substitution is, indeed, called for by the character of the differential 
equation. 1 Equation 17-19 is written in the standard form 

d*F dF 

^p + Pte^ + tfer-o. 

d*F 
the coefficient of — — being unity. The coefficients p and q in Equation 
dp 

17-19 possess singularities 2 at £ = 0. The singular point £ = is a regular 

point, however, inasmuch as p(£) is of order l/£ and q(£) of order l/£ 2 . To 

solve a differential equation possessing a regular point at the origin, the 

substitution 17-20 is made in general. It is found that it leads to an indidal 

equation from which the index s can be determined. 

Since we are interested only in acceptable wave functions, we shall ignore 

negative values of s. For this reason we could assume F(£) to contain only 

positive powers of £. Occasionally, however, the indicial equation leads to 

non-integral values of s, in which case the treatment is greatly simplified by 

the substitution 17-20. 

If we introduce the series 17-20 into Equation 17-19 and group 
together coefficients of equal powers of £ , we obtain the equation 

(s 2 - m 2 )a e~ 2 + {(* + l) 2 - w^oif- 1 
+ 



+ 



{(« + 2) 2 - m 2 \a 2 + \~~ 2( 5 + l)|a ]f + • • • 
{(« + v) 2 - m 2 )a v + <X - 2(« + v - 1)L_ 2 1^" 



+ . . . =o. (17-21) 



Since this is an identity in £ , that is, an equation which is true 
for all values of £ , we can show that the coefficient of each power 

1 See the standard treatments of the theory of linear differential equations; 
for example, Whittaker and Watson, "Modern Analysis," Chap. X. 

* A singularity for a function p(£) is a point at which p(£) becomes infinite. 



110 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV-17 

of £ must be itself equal to zero. This argument gives the set of 
equations 

(s 2 - m 2 )a = 0, (17-22a) 

{(« + l) 2 - m 2 }^ = 0, (17-226) 



(17-22c) 



{(s + v) 2 - m*}a v + $L - 2(* + v - 1) ja,_ 2 = 0, 

etc. 

The first of these, 17-22a, is the indicial equation. From it 
we see that s is equal to +m or — ra, inasmuch as a is not equal 
to zero. In order to obtain a solution of the form of Equation 
17-20 which is finite at the origin, we must have s positive, so 
that we choose s = +\m\. This value of s inserted in Equation 
17-226 leads to the conclusion that a,i must be zero. Since the 
general recursion relation 17-22c connects coefficients whose 
subscripts differ by two, and since a\ is zero, all odd coefficients 
are zero. The even coefficients may be obtained in terms of 
a by the use of 17-22c. 

However, just as in the case of the linear harmonic oscillator, 
the infinite series so obtained is not a satisfactory wave function 
for general values of X', because its value increases so rapidly with 
increasing £ as to cause the total wave function to become 
infinite as £ increases without limit. In order to secure an 
acceptable wave function it is necessary to cause the series to 
break off after a finite number of terms. The condition that the 
series break off at the term a n 'Z n ' Hm] , where n f is an even integer, 
is obtained from 17-22c by putting n' + 2 in place of v and equat- 
ing the coefficient of a n , to zero. This yields the result 

V = 2(|m| -h n! + 1)«. (17-23) 

Combining the expressions for \ z and X' given by Equations 
17-11 and 17-23, we obtain the result 

X = V + X, = 2(|m| + ri + l)a + 2(n, + H)a z , (17-24) 

or, on insertion of the expressions for X, a, and a z , 

W mn >n t = (H + n' + l)hv + (n. + \Qhv t . (17-25) 

In the case of the isotropic harmonic oscillator, with v z = vo» 
this becomes 



IV-17] THREE-DIMENSIONAL HARMONIC OSCILLATOR 111 

W n = (n + y 2 )hv Qi n = \m\+n' + n.. (17-26) 
The quantum numbers are restricted as follows: 

m = 0, ±1, ±2, • • • , 
n' = 0, 2, 4, 6, • • , 
n* = 0, 1, 2, • • • . 

These lead to the same quantum weights for the energy levels 
as found in Section 15. 
The wave functions have the form 

$n>mn-(p, <P, z) = Ne^e r F |m , ,»' ( V«p)e 2 * H n ,(V<**), 

(17-27) 

in which JV is the normalization constant and F, m ,, n >(\/ap) is a 
polynomial in p obtained from Equation 17-20 by the use of the 
recursion relations 17-22 for the coefficients a v . "It contains only 
odd powers of p if \m\ is odd, and only even powers if \m\ is 
even. 

Problem 17-1. The equation for the free particle is separable in many 
coordinate systems. Using cylindrical polar coordinates, set up and 
separate the wave equation, obtain the solutions in <p and z, and obtain the 
recursion formula for the coefficients in the series solution of the p equation. 
Hint: In applying the polynomial method, omit the step of finding the 
asymptotic solution. 

Problem 17-2. Calculate pi for a harmonic oscillator in a state repre- 
sented by ypn'mn, of Equation 17-27. Shew that p x is zero in the same state. 
Hint: Transform — into cylindrical polar coordinates. 

Problem 17-3. The equation for the isotropic harmonic oscillator is 
separable also in spherical polar coordinates. Set up the equation in these 
coordinates and carry out the separation of variables, obtaining the three 
total differential equations. 



CHAPTER V 

THE HYDROGEN ATOM 

The problem of the structure of the hydrogen atom is the most 
important problem in the field of atomic and molecular structure, 
not only because the theoretical treatment of this atom is simpler 
than that of other atoms and of molecules, but also because it 
forms the basis for the discussion of more complex atomic sys- 
tems. The wave-mechanical treatment of polyelectronic atoms 
and of molecules is usually closely related in procedure to that 
of the hydrogen atom, often being based on the use of hydrogen- 
like or closely related wave functions. Moreover, almost without 
exception the applications of qualitative and semiquantitative 
wave-mechanical arguments to chemistry involve the functions 
which occur in the treatment of the hydrogen atom. 

The hydrogen atom has held a prominent place in the develop- 
ment of physical theory. The first spectral series expressed by a 
simple formula was the Balmer series of hydrogen. Bohr's 
treatment of the hydrogen atom marked the beginning of the old 
quantum theory of atomic structure, and wave mechanics had 
its inception in Schrodinger's first paper, in which he gave the 
solution of the wave equation for the hydrogen atom. Only 
in Heisenberg's quantum mechanics was there extensive develop- 
ment of the theory (by Heisenberg, Born, and Jordan) before 
the treatment of the hydrogen atom, characterized by its diffi- 
culty, was finally given by Pauli. In later developments, beyond 
the scope of this book, the hydrogen atom retains its important 
position; Dirac's relativistic quantum theory of the electron 
is applicable only to one-electron systems, its extension to 
more complicated systems not yet having been made. 

The discussion of the hydrogen atom given in this chapter is 
due to Sommerfeld, differing in certain minor details from that 
of Schrddinger. It is divided into four sections. In the first, 
Section 18, the wave equation is separated and solved by the 
polynomial method, and the energy levels are discussed. Sec- 

112 



V-18a] THE SOLUTION OF THE WAVE EQUATION 113 

tions 19 and 20 include the definition of certain functions, the 
Legendre and Laguerre functions, which occur in the hydrogen- 
atom wave functions, and the discussion of their properties. A 
detailed description of the wave functions themselves is given 
in Section 21. 

18. THE SOLUTION OF THE WAVE EQUATION BY THE POLY- 

NOMIAL METHOD AND THE DETERMINATION OF 

THE ENERGY LEVELS 

18a. The Separation of the Wave Equation. The Transla- 

tional Motion. — We consider the hydrogen atom as a system of 

two interacting point particles, the interaction being that due 

to the Coulomb attraction of their electrical charges. Let us 

for generality ascribe to the nucleus the charge +Ze, the charge 

of the electron being —e. The potential energy of the system, 

Ze 2 

in the absence of external fields, is > in which r is the distance 

r 

between the electron and the nucleus. 

If we write for the Cartesian coordinates of the nucleus and 

the electron x\ } yi, Z\ and x^ y^ £2, and for their masses mi and 

ra 2 , respectively, the wave equation has the form 

1 (d 2 ^ <fth ayA , ±_(<*ti . *Vj . a yA 

m\ dx\ + dy\ + dz\ ) + m\ dxl + dy\ + dz\ ) 

fcn-2 

+ jfiWr ~ V)+t = 0, (18-1) 



in which 

V = - 



Ze 2 



V(X2 - *i) 2 + (2/2 - Vi) 2 + (Z2 - zi) 2 



Here the subscript T (signifying total) is written for W and ^ to 
indicate that these quantities refer to the complete system, with 
six coordinates. 

This equation can be immediately separated into two, one of 
which represents the translational motion of the molecule as a 
whole and the other the relative motion of the two particles. 
In fact, this separation can be accomplished in a somewhat more 
general case, namely, when the potential energy V is a general 
function of the relative positions of the two particles, that is, 
V = V(x*i — x\ 9 2/2 — yi, z*k — 21). This includes, for example, 
the hydrogen atom in a constant electric field, the potential 



114 THE HYDROGEN ATOM [V-18a 

energy due to the field then being eEz* — eEzi = eE(z* — Z\) f 
in which E is the strength of the field, considered as being in the 
direction of the z axis. 

To effect the separation, we introduce the new variables 
x, t/, and z, which are the Cartesian coordinates of the center of 
mass of the system, and r, #, and <p, the polar coordinates of the 
second particle relative to the first. These coordinates are 
related to the Cartesian coordinates of the two particles by the 
equations 



X - . > 

rrii + m 2 


(18-2a) 


rriiyi + m 2 y 2 

y — , 

mi + m 2 


(18-26) 


miZi + ra 2 2 2 
z = . y 
mi + m 2 


(18-2c) 


r sin # cos <p = x 2 — xi, 


(18-2d) 


r sin & sin ^> = y 2 — yi, 


(18-26) 


r cos t? = 2 2 — 2i. 


(18-2/) 



The introduction of these new independent variables in 
Equation 18-1 is easily made in the usual way. The resultant 
wave equation is 

l / ay r ay r ay r \ in_ ^ / ,d*A 

mi + m 2 \^ ax 2 "*" dy 2 + a* 2 j "*" M |r 2 dry ar / 

l aVr 



j : — r_m + 

• ~2 «i^2 JH 3.-2 » 



(**£)} 



r 2 sin 2 # d<p 2 ' r 2 sin # d& 

&7T 2 

+ ^- { W T - V(r, », <p)\+t= 0. (18-3) 

In this equation the symbol /x has been introduced to represent 
the quantity 



raira 2 



mi + m 2 



fori = — +— V (18-4) 

y ju mi m 2 / 



M is the reduced mass of the system, already discussed in Section 
2d in the classical treatment of this problem. 

It will be noticed that the quantity in the first set of parentheses 
is the Laplacian of \f/ T in the Cartesian coordinates x, y } and 2, 
and the quantity in the first set of braces is the Laplacian in the 
polar coordinates r, #, and <p (Appendix IV). 



V-18a] THE SOLUTION OF THE WAVE EQUATION 115 

We now attempt to separate this equation by expressing xpr 
as the product of a function of x> y, z and a function of 
r, #, <p, writing 

iMx, y, z, r, tf, <p) = F(z, y, z)+(r, *, *). (18-5) 

On introducing this in Equation 18-3 and dividing through by 
\p T = Fyf/j it is found that the equation is the sum of two parts, 
one of which is dependent only on x f y, and z and the other 
only on r, #, and (p. Each part must hence be equal to a con- 
stant. The resulting equations are 

d 2 F d 2 F d 2 F 8 T 2( mi +m 2 ) 

W + W + W + h 2 WtrF = °> (18_6) 

and 

r 2 dry dr)"^ r 2 sin 2 t> <V "*" r 2 sin d0^ Sin d#/ 

+ ^{Tf - 7(r, #,*)}*=(>, (18-7) 

with 

TT , r + w = Wt. (18-8) 

Equation 18-6 is identical with Equation 13-2 of Section 13, 
representing the motion of a free particle; hence the translational 
motion of the system is the same as that of a particle with mass 
mi + m 2 equal to the sum of the masses of the two particles. 
In most problems the state of translational motion is not impor- 
tant, and a knowledge of the translational energy W tr is not 
required. In our further discussion we shall refer to W, the 
energy of the system aside from the translational energy, simply 
as the energy ctf the system. 

Equation 18-7 is identical with the wave equation of a single 
particle of mass m under the influence of a potential function. 
V(r f #, if) . This identity corresponds to the classical identity of 
Section 2d (Eqs. 2-25). 

If we now restrict ourselves to the case in which the potential 
function V is a function of r alone, 

V = V(r), 

Equation 18-7 can be further separated. We write 

f (r, *, <p) = R(r) ■ 9(#) • #(*) ; (18-9) 



116 THE HYDROGEN ATOM [V-18a 

on introducing this in Equation 18-7 and dividing by RQ$, it 
becomes 

JL il 2^\ . 1 d 2 * 1 dj . d§\ 

r 2 R dr\ dr) + > 2 sin 2 #<*> d<p 2 "*" r 2 sin #0 d#\ Sm dt>/ 

+ %r\w- V(r)} -o. (18-10) 

On multiplying through by r 2 sin 2 #, the remaining part of the 

1 d 2 $ 
second term, — ^—z, which could only be a function of the inde- 
<f> d<p 2 

pendent variable <p, is seen to be equal to terms independent of <p. 

Hence this term must be equal to a constant, which we call — m 2 : 

d 2 $ 

The equation in & and r then can be written as 

1 l( r ^\ _ J** _L_ A( 8in <M + §^ 
R dr\ dr / sin 2 ^ sin dQ d&\ d#J ^ h 

{W - V(r)\ = 0. 

The part of this equation containing the second and third terms 
is independent of r and the remaining part is independent of #, 
so that we can equate each to a constant. If we set the # terms 
equal to the constant — 0, and the r terms equal to +fi f we 
obtain the following equations, after multiplication by 9 and 
by R/r 2 , respectively: 



8irV 2 

2 



and 






Equations 18-11, 18-12, and 18-13 are now to be solved 
in order to determine the allowed values of the energy. 
The sequence of solution is the following: We first find that 
Equation 18-11 possesses acceptable solutions only for certain 
values of the parameter m. Introducing these in Equation 18-12, 
we find that it then possesses acceptable solutions only for 
certain values of 0. Finally, we introduce these values of 



V-18bl THE SOLUTION OF THE WAVE EQUATION 117 

in Equation 18-13 and find that this equation then possesses 
acceptable solutions only for certain values of W. These are the 
values of the energy for the stationary states of the system. 

It may be mentioned that the wave equation for the hydrogen 
atom can also be separated in coordinate systems other than the 
polar coordinates r, #, and <p which we have chosen, and for some 
purposes another coordinate system may be especially appro- 
priate, as, for example, in the treatment of the Stark effect, 
for which (as shown by Schrodinger in his third paper) it is 
convenient to use parabolic coordinates. 

18b. The Solution of the <p Equation. — As was discussed in 
Section 17, the solutions of Equation 18-11, involving the cyclic 
coordinate <p, are 

<M*>) = -*=^*. (18-14) 

In order for the function to be single-valued at the point <p — 
(which is identical with <p = 2tt)> the parameter m must be equal 
to an integer. The independent acceptable solutions of the (p 
equation are hence given by Equation 18-14, with m = 0, 
+ 1, +2, • • • , — 1, ^-2, • • k ,* these values are usually 
written as 0, ±1, ±2, • • • , it being understood that positive 
and negative values correspond to distinct solutions. 

The constant m is called the magnetic quantum number. It is 
the analogue of the same quantum number in the old-quantum- 
theory treatment (Sec. 76). 

The factor 1/\/2t is introduced in order to normalize the 
functions $m(<p), which then satisfy the equation 

J%*M<M*>)^ = 1. (18-15) 

It may be pointed out that for a given value of \m\ (the 
absolute value of ra), the two functions $\ m \(<p) and 3>_ w (¥?) 
satisfy the same differential equation, with the same value of the 
parameter, and that any linear combination of them also satisfies 
the equation. The sum and the difference of these two functions 
are the cosine and sine functions. It is sometimes convenient 
to use these in place of the complex exponential functions as the 
independent solutions of the wave equation, the normalized 
solutions then being 



118 THE HYDROGEN ATOM [V-18c 

1 



$o(v>) = 

V3ftr 
/ 1 

cos \m\ <p, \ (18-16) 



$\m\(<p) = 



V2t 

\-\Zw 

— /-sin \m\<p, \m\ = 1,2,3, • • 



There is only one solution for \m\ = 0. These functions are 
normalized and are mutually orthogonal. 

It is sometimes convenient to use the symbol m to represent 
the absolute value of the magnetic quantum number as well 
as the quantum number itself. To avoid confusion, however, 
we shall not adopt this practice but shall write |ra| for the 
absolute value of m. 

18c. The Solution of the # Equation. — In order to solve the 
& equation 18-12, it is convenient for us to introduce the new 
independent variable 

z = cos tf, (18-17) 

which varies between the limits — 1 and +1, and at the same time 
to replace 9(#) by the function P(z) to which it is equal: 

P(z) = 6(#). (18-18) 

Noting that sin 2 # = 1 — z 2 and that 

d9 dPdz^ _dP . 
d* dz dd dz Sm ' 

we see that our equation becomes 

On attempting to solve this equation by the polynomial method, 
it is found that the recursion formula involves more than two 
terms. If, however, a suitable substitution is made, the equa- 
tion can be reduced to one to which the polynomial method can 
be applied. 

The equation has singular points at z = ±1, both of which are regular 
points (see Sec. 17), so that it is necessary to discuss the indicial equation 
at each of these points. In order to study the behavior near z — -f-1, it L? 



V-18c] THE SOLUTION OF THE WAVE EQUATION 119 

convenient to make the substitution x - 1 — z, R(x) = P(z), bringing this 
point to the origin of x. The resulting equation is 

00 

If we substitute R — x* ^X a^x" in this equation, we find that the indicial 

equation (see Sec. 17) leads to the value \m\/2 for s. Likewise, if we investi- 
gate the point z = —1 by making the substitution y = 1 -}- z and similarly 
study the indicial equation at the origin of y, we find the same value for the 
index there. 

The result of these considerations is that the substitution 

I ml 1ml I ml 

P(z) = x2y2G(z) = (1 - z 2 ) YG(z) (18-20) 

is required. On introducing this into Equation 18-19, the differ- 
ential equation satisfied by G(z) — which should now be directly 
soluble by a power series — is found to be 

(1 - z 2 )G" - 2(|m| + l)zG' + 

[p - \m\(\m\ + 1))0 = 0, (18-21) 

in which G' represents -r- and G" represents -rj- 

This equation we now treat by the polynomial method, the 
successive steps being similar to those taken in Section 11 in the 
discussion of the harmonic oscillator. Let 

G = a + aiz + a 2 z 2 + a z z 3 + • • • , (18-22) 

with G' and G" similar series obtained from this by differentiation. 
On the introduction of these in Equation 18-21, it becomes 

1 • 2a 2 + 2 • 3a 3 z + 3 • 4a 4 z 2 + 4 • 5a b z* + • • • 
- 1 • 2a 2 z 2 - 2 • 3a 3 z 3 - • • • 
-2(|m| + l)aiz -2 • 2(|m| + l)a 2 z 2 -2 • 3(|m| + l)a 3 z 3 - • • • 
+ {P - |m|(|m| + l)}a + {}a l2 + {}a 2 z 2 + {}a 3 z 3 + • • • = 0, 
in which the braces {} represent {/3 — |m|(|m| + 1)}. This 
equation is an identity in z, and hence the coefficients of indi- 
vidual powers of z must vanish; that is, 

1 • 2a 2 + {}a = 0, 

2-3a 3 + ({) - 2(|m| + l)) fll =0, 

3 • 4a 4 + ({ } - 2 • 2(|m| + 1) - 1 . 2)a 2 = 0, 

4 • 5a 6 + ({ } - 2 • 3(|m| + 1) - 2 • 3)a 8 = 0, 



120 THE HYDROGEN ATOM [V-18c 

or, in general, for the coefficient of z", 

(v + 1)(» + 2K+ 2 + UP - \m\(\m\ + 1)} 

- 2v(\m\ + 1) - v(v - 1)K = 0. 

This leads to the two-term recursion formula 



_ JL± H)(" + H + 1 ) -0 „ 
(„ + !)(„ + 2) 



(18-23) 



between the coefficients a„ +2 and a, in the series for (?. 

It is found on discussion by the usual methods 1 that an infinite 
series with this relation between alternate coefficients converges 
(for any values of \m\ and p) for — 1 < z < 1, but diverges for 
z = +1 or — 1, and in consequence does not correspond to an 
acceptable wave function. In order to be satisfactory, then, 
our series for G must contain only a finite number of terms. 
Either the even or the odd series can be broken off at the term 
in z v ' by placing 

P = (/ + \m\)(v' + \m\ + 1), / = 0, 1, 2, • • • , 

and the other series can be made to vanish by equating a\ or a 
to zero. The characteristic values of the parameter p are thus 
found to be given by the above expression, the corresponding 
functions G(z) containing only even or odd powers of z as / 
is even or odd. 

It is convenient to introduce the new quantum number 

I = v' + \m\ (18-24) 

in place of /, the allowed values for I being (from its definition) 
|m|, \m\ + 1, |m| + 2, • • • . The characteristic values of p 
are then 

P = 1(1 + 1), I = (m|, \m\ + 1, ' * ' . (18-25) 

I is called the azimuthal quantum number; it is analogous to the 
quantum number k of the old quantum theory. Spectral states 
which are now represented by a given value of I were formerly 
represented by a value of k one unit greater, k = 1 corresponding 
to I = 0, and so on. 

1 R. Courant and D. Hilbert, " Methoden der mathematischen Physik," 
2d ed.,Vol. I, p. 281, Julius Springer, Berlin, 1931. 



V-18d] 



THE SOLUTION OF THE WAVE EQUATION 



121 



We have now shown that the allowed solutions of the # equa* 
tion are 9(#) = (1 - 2 2 ) lm|/2 G(z), in which G(z) is denned by 
the recursion formula 18-23, with p = 1(1 + 1). It will be shown 
in Section 19 that the functions G(#) are the associated Legendre 
functions. A description of the functions will be given in 
Section 21. 

18d. The Solution of the r Equation. — Having evaluated /? as 
1(1 + 1), the equation in r becomes 



i d( dR\ r Ki + D 

r 2 dr\ dr J ^ |_ r 2 ^ 



S ^ {W _ v(r)\ 



R = 0, (18-26) 



in which V(r) = — Ze 2 /r, Z being the atomic number of the atom. 
It is only now, by the introduction of this expression for the 
potential energy, that we specialize the problem to that of the 
one-electron or hydrogenlike atom. The discussion up to this 
point is applicable to any system of two particles which interact 
with one another in a way expressible by a potential function 
V(r)j as, for example, the two nuclei in a diatomic molecule after 
the electronic interactions have been considered by the Born- 
Oppenheimer method (Sec. 35a). 

Let us first consider the case of W negative, corresponding to 
a total energy insufficient to ionize the atom. Introducing the 
symbols 



and 



8t 2 ixW \ 
h 2 

±T 2 >iZe 2 
h 2 a ' 



(18-27) 



and the new independent variable 

p = 2ar, 
the wave equation becomes 
1 d( dS\ ( 1 1(1 + i) , x\ 



(18-28) 



^ p ^ » , (18-29) 



122 THE HYDROGEN ATOM [V-lSd 

in which S(p) = R (r). As in the treatment of the harmonic 
oscillator, we first discuss the asymptotic equation. For p 
large, the equation approaches the form 



dp 
the solutions of which are 



d*S l e 



S = e^ and 8 = e \ 



Only the second of these is satisfactory as a wave function. 

We now assume that the solution of the complete equation 
18-29 has the form 



p 



S(p) = e 2 F(p). (18-30) 

The equation satisfied by F(p) is found to be 

^ p ^ oo. (18-31) 

The coefficients of F' and F possess singularities at the origin, 
which is a regular point (cf. Sec. 17), so that we again make the 
substitution 

F(p) = P 'L(p), (18-32) 

in which L(p) is a power series in p beginning with a non- vanishing 
constant term: 

L(p) = %a»P v , a 5* 0. (18-33) 

V 

Since 

F'(p) = sp'-'L + p'U 
and 

F"( P ) = s(s - 1) P - 2 L + 2s P - 1 L' + p'L", 

Equation 18-31 becomes 

p-+*L" + 2sp'+ 1 L' + s(s - l)p'L 
+ 2p'+ l L' + 2sp'L 
- p«+ 2 L' - sp'+ l L 
+ (X - 1) P '+>L - 1(1 + l)p'L = 0. (18-34) 



V-18d] THE SOLUTION OF THE WAVE EQUATION 123 

Since L begins with the term a , the coefficient of p* is seen to be 
{s(s — 1) + 2s — 1(1 + l)}a , and, since a does not vanish, 
the expression in braces must vanish in order for Equation 18-34 
to be satisfied as an identity in p. This gives as the indicial 
equation for s: 

s(s + 1) - 1(1 + 1) = 0, or 5 = +1 or -(1 + 1). (18-35) 

Of the two solutions of the indicial equation, the solution 
s = — (J + 1) does not lead to an acceptable wave function. 
We accordingly write 

F(p) = p l L(p), (18-36) 

and obtain from 18-34 the equation 

P L" + {2(1 + 1) - p}Z/ + (X - I - 1)L = 0, (18-37) 
after substituting I for s and dividing by p l+l . We now introduce 
the series 18-33 for L in this equation and obtain an equation 
involving powers of p, the coefficients of which must vanish 
individually. These conditions are successively 

(X - I - l)a + 2(1 + l) ai = 0, 
(X — I — 1 — IK + {2 . 2(i + 1) + 1 • 2}a 2 = 0, 
(X - I - 1 - 2)a 2 + {3-2(Z + 1) +2-3}a 3 = 

or, for the coefficient of p", 

(X — « — 1 — 0a„ + {2(x + 1)(I + 1) + K^ + l)}a, +1 = 0. 

(18-38) 

It can be shown by an argument similar to that used in Section 
11a for the harmonic oscillator that for any values of X and I the 
series whose coefficients are determined by this formula leads to 
a function S(p) unacceptable as a wave function unless it breaks 
off. For very large values of v the successive terms of an infinite 
series given by 18-38 approach the terms of the expansion 
of e p , which accordingly represents the asymptotic behavior of 
the series. This corresponds to an asymptotic behavior of 

S(p) = e 2 p l L(p) similar to e 2 , leading to the infinity catastrophe 
with increasing p. 

Consequently the series must break off after a finite number 
of terms. The condition that it break off after the term in p n ' is 
seen from Equation 18-38 to be 



124 THE HYDROGEN ATOM [V-18e 

X-Z-l-n'=0 
or 

X = n, where n = n' + I + 1. (18-39) 

n' is called the radial quantum number and n the fofoZ quantum 
number. From its nature it is seen that n' can assume the values 
0, 1, 2, 3, • • • . The values of n will be discussed in the next 
section. 

In this section we have found the allowed solutions of the 

r equation to have the form R(r) = e 2 p l L(p), in which L(p) is 
defined by the recursion formula 18-38, with X = n. It will 
be shown in Section 20 that these functions are certain associated 
Laguerre functions, and a description of them will .be given in 
Section 21. 

18e. The Energy Levels. — Introducing for X its value as given 
in Equation 18-27, and solving for W y it is found that Equation 
18-39 leads to the energy expression 

Wn = - W = _**£' = - Z 1 Wh> (18-40) 
h*n 2 n 2 n 2 

in which 

R = W£* and W H = Rhc. 
h 3 c 

This expression is identical with that of the old quantum theory 
(Eq. 7-24), even to the inclusion of the reduced mass ju- It is 
seen that the energy of a hydrogenlike atom in the state repre- 
sented by the quantum numbers n', l } and m does not depend on 
their individual values but only on the value of the total quantum 
number n = n' + I + 1. Inasmuch as both n' and I by their 
nature can assume the values 0, ,1, 2, • • • , we see that the 
allowed values of n are 1, 2, 3, 4, • • • , as assumed in the old 
quantum theory and verified by experiment (discussed in 
Sec. 76). 

Except for n = 1, each energy level is degenerate, being 
represented by more than one independent solution of the wave 
equation. If we introduce the quantum numbers n, l } and m 
as subscripts (using n in preference to n') y the wave functions 
we have found as acceptable solutions of the wave equation 
may be written as 

*nJm(r, 0, <p) = flni(r)ei m (tf)* w (*), (18-41) 



V-19] LEGENDRE FUNCTIONS AND SURFACE HARMONICS 125 

the functions themselves being those determined in Sections 
186, 18c, and 18d. The wave functions corresponding to distinct 
sets of values for n, Z, and m are independent. The allowed 
values of these quantum numbers we have determined to be 

m = 0, ±1, ±2, • • • , 
I = \m\, \m\ + 1, \m\ + 2, • • • , 
n = I + 1, I + 2, I + 3, • • • . 

This we may rewrite as 

total quantum number n — 1, 2, 3, • • • , 
azimuthal quantum number Z = 0, 1, 2, • • • , n — 1, 
magnetic quantum number m = — Z, --Z + 1, • • • ,—1, 0, 

+ 1, • • • , +Z - 1, +Z. 

There are consequently 21 + 1 independent wave functions with 
given values of n and Z, and n 2 independent wave functions with 
a given value of n, that is, with the same energy value. The 
2Z + 1 wave functions with the same n and Z are said to form a 
completed subgroup, and the n 2 wave functions with the same n a 
completed group. The wave functions will be described in the 
following sections of this chapter. 

A similar treatment applied to the wave equation with W 
positive leads to the result that there exist acceptable solutions 
for all positive values of the energy, as indicated by the general 
discussion of Section 9c. It is a particularly pleasing feature of 
the quantum mechanics that a unified treatment can be given 
the continuous as well as the discrete spectrum of energy values. 
Because of the rather complicated nature of the discussion of the 
wave functions for the continuous spectrum (in particular their 
orthogonality and normalization properties) and of their minor 
importance for most chemical problems, we shall not treat them 
further. 1 

19. LEGENDRE FUNCTIONS AND SURFACE HARMONICS 

The functions of & which we have obtained by solution of the 
# equation are well known to mathematicians under the name of 
associated Legendre functions. 2 The functions of d and <p are 

1 See Sommerfeld, " Wave Mechanics," p. 290. 

2 The functions of # for m =0 are called Legendre functions. The asso- 
ciated Legendre functions include the Legendre functions and additional 



126 THE HYDROGEN ATOM [V-lOa 

called surface harmonics (or, in case cosine and sine functions of 
<p are used instead of exponential functions, tesseral harmonics). 
We could, of course, proceed to develop the properties of these 
functions from the recursion formulas for the coefficients in the 
polynomials obtained in the foregoing treatment. This would 
be awkward and laborious, however; it is simpler for us to define 
the functions anew by means of differential expressions or 
generating functions, and to discuss their properties on this basis, 
ultimately proving the identity of these functions with those 
obtained earlier by application of the polynomial method. 

19a. The Legendre Functions or Legendre Polynomials. — The 
Legendre functions or Legendre polynomials Pj(cos #) = Pi(z) 
may be defined by means of a generating function T(t, z) such 
that 

r( M)-^X- vl _' 2te + 1 , OH) 

As in the case of the Hermite polynomials (Sec. lie), we 
obtain relations among the polynomials and their derivatives by 
differentiating the generating function with respect to t and to z. 
Thus on differentiation with respect to t, we write 

00 



-S"*""--^ 



dt ^-J ' (l-2zt + t 2 )* 

J~0 

or 

(1 - 2zt + t^lPit 1 - 1 ^ (z - t)%P t t l 
i i 

(the right side having been transformed with the use of Equation 
19-1), and consequently, by equating coefficients of given powers 
of t on the two sides, we obtain the , recursion formula for the 
Legendre polynomials 

(I + l)Pi+iC0 ~ (21 + l)zPi(z) + IPi-^z) = 0. (19-2) 

Similarly, by differentiation with respect to z, there is obtained 

dz ~~ 2f l ~ (1 ~ 2zt + t 2 )* 
i 

functions (corresponding to I ml > 0) conveniently defined in terms of the 
Legendre functions. 



V-19b] LEGENDRE FUNCTIONS AND SURFACE HARMONICS 127 
or 

(1 -2zt + t*)2 f P' l t' = t2 f P l t', 

i i 

which gives the relation 

PUz) - 2zP[(z) + P'Uz) - P t (z) = (19-3) 

involving the derivatives of the polynomials. Somewhat simpler 
relations may be obtained by combining these. From 19-2 
and 19-3, after differentiating the former, we find 

zP[(z) - P'^z) ~ IPi(z) = (19-4) 

and 

P[ +l {z) - zP\(z) - (1 + l)P t (z) = 0. (19-5) 

We can now easily find the differential equation which Pi{z) 
satisfies. Reducing the subscript Z to Z — 1 in 19-5, and sub- 
tracting 19-4 after multiplication by z, we obtain 

(i - * 2 )p; + up 1 - zp,_! = o, 

which on differentiation becomes 

Tz{ (l ~ z2)d -^r\ + lPl{z) + lzP ' i{z) ~ lp '- i{z) = °- 

The terms in P\ and P[_i may be replaced by l 2 P t , from 19-4, 
and there then results the differential equation for the Legendre 
polynomials 

~{(1 ~ * 2 )~^} + 1(1 + DftW = 0. (19-6) 

19b. The Associated Legendre Functions. — We define the 
associated Legendre functions of degree Z and order \m\ (with 
values I = 0, 1, 2, • • • and \m\ = 0, 1, 2, • • • , Z) in terms of 
the Legendre polynomials by means of the equation 

PlT'CO - (1 - z 2 )' ml/2 |S p '(z). (19-7) 

[It is to be noted that the order \m\ is restricted to positive values 
(and zero) ; we are using the rather clumsy symbol \m\ to represent 
the order of the associated Legendre function so that we may 
later identify m with the magnetic quantum number previously 



128 THE HYDROGEN ATOM [V-19b 

introduced.] The differential equation satisfied by these func- 
tions may be found in the following way. On differentiating 
Equation 19-6 \m\ times, there results 

rf\m\+2p ( z \ d^ m ^ l Pi(z\ 

(i - * 2 )^p£p " 2(M + i)^-^£P 

+ {l(l + l)- |m|(H + DI^P = (19-8) 

d lml Pi(z) 
as the differential equation satisfied by — |m , • With the 

use of Equation 19-7 this equation is easily transformed into 

_ *£%p _ »£«■> + {«, + „ - J±Jjpro) 

= 0, (19-9) 

which is the differential equation satisfied by the associated 
Legendre function P\ ml (z). 

This result enables us to identify 1 the & functions of Section 
18c (except for constant factors) with the associated Legendre 
functions, inasmuch as Equation 19-9 is identical with Equation 
18-19, except that P(z) is replaced by P\ ml (z) and fi is replaced 
by 1(1 + 1), which was found in Section 18c to represent the 
characteristic values of £. Hence the wave functions in # 
corresponding to given values of the azimuthal quantum number I 
and the magnetic quantum number m are the associated Legendre 
functions P[ ml (z). 

The associated Legendre functions are most easily tabulated by 
the use of the recursion formula 19-2 and the definition J 9-7, 
together with the value Pq{z) = 1 as the starting point. A 
detailed discussion of the functions is given in Section 21. 

For some purposes the generating 'function for the associated 
Legendre functions is useful. It is found from that for the 
Legendre polynomials to be 

T, ,(z t) = ^?P™(z)t< = (ZMWl-**) 1 - 1 ^ 1 " 1 
J|«i^» l > ~ ^n W* - 2i»i(|m|)!(l - 2zt + «*)i-"+M 
i-M 

(19-10) 

1 The identification is completed by the fact that both functions are formed 
from polynomials of the same degree. 



V-20a] THE LAGUERRE POLYNOMIALS 129 

In Appendix VI it is shown that 



(19-11) 



c +i (ofoi-r ?*i, 

FTKzWWdz = { 2 (l + \m\)\ _ 

Using this result, we obtain the constant necessary to normalize 
the part of the wave function which depends on &. The final 
form for 0(#) is 



e W = ^?L^> <^M>i p , W) . (19 _ 12) 

Problem 19-1. Prove that the definition of the Legendre polynomials 

Poto - 1, ) 

""■"-ar^ '->•*••■ I (,M " 

is equivalent to that of Equation 19-1. 

Problem 19-2. Derive the following relations involving the associated 
Legendre functions: 

(i - *' W-'(*) = ^j-fjM+'iM - ^pTj pl "i(*), (19 ~ 14 > 
(i - «w« w - « + H^ (l + + W + % ) - 

a-H^-H^)^, (19 _ 15) 

and 

^' (Z) (2? + 1) Pl - l{2) + (21 + 1) Pl+M ' (1 ^ 16) 

20. THE LAGUERRE POLYNOMIALS AND ASSOCIATED LAGUERRE 

FUNCTIONS 

20a. The Laguerre Polynomials. — The Laguerre polynomials 
of a variable p, within the limits ^ p ^ oo , may be defined by 
means of the generating function 

^«)-2^ r -r^- (2(M) 

r=0 

To find the differential equation satisfied by these polynomials 
Lr(p), we follow the now familiar procedure of differentiating the 



130 THE HYDROGEN ATOM [V-20a 

dU 
generating function with respect to u and to p. From — we 

obtain 

2 Lr(o) t = e i-y _P_ _ P" , _J_\ 
(r - 1) ! 1 - u\ 1 - « (1 - w) 2 ^ 1 - u/ 



or 

r r 

from which there results the recursion formula 

L r+l (p) + ( p - 1 - 2r)L r ( P ) + r 2 L r _!( P ) = 0. (20-2) 

Similarly from — we have 

^i r! 1 — Uj^j r\ 

r r 

or 

L' r (p) - ^r'-i(p) + rLr-i(p) = 0, (20-3) 

in which the prime denotes the derivative with respect to p. 
Equation 20-3 may be rewritten and differentiated, giving 

KM =(r + l){L;(p) -L r (p)\ 
and 

L' r ' +l (p) =(r + l){L;'(p)-L;(p)l, 

with similar equations for L' r+2 (p) and L" +2 (p). Replacing r by 
r + 1 in Equation 20-2 and differentiating twice, we obtain 
the equation 

KM + (P ~ 3 - 2r)L£ 1 (p) + (r + l) 2 L^(p) + 2L r ' +1 (p) = 0. 

With the aid of the foregoing expressions this is then transformed 
into an equation in L r (p) alone, 

pL'/ip) + (1 - p)L' r {p) + rL r (p) = 0, (20-4) 

which is the differential equation for the rth Laguerre polynomial. 

d r 
Problem 20-1. Show that L f (p) = e p —(p r e" p ). 

dp r 



V-SM)b] THE LAGUERRE POLYNOMIALS 131 

20b. The Associated Laguerre Polynomials and Functions.— 

The sth derivative of the rth Laguerre polynomial is called the 
associated Laguerre polynomial of degree r — s and order s: 

L' r ( P ) = £-Mp)- (20-6) 

The differential equation satisfied by L' r (p) is found by differ- 
entiating Equation 20-4 to be 

pL' r "{ P ) + (s + 1 - p)L' r \p) + (r - s)L< r {p) = 0. (20-6) 

If we now replace r by n + I and s by 21 + 1, Equation 20-6 
becomes 

pL^V'Xp) + {2(J + 1) - p)L^\p) 

+ (n - i - l)L 2 n VV(p) = 0. (20-7) 

On comparing this with Equation 18-37 obtained in the treat- 
ment of the r equation for the hydrogen atom by the polynomial 
method, we see that the two equations are identical when 
Ln+i l (p) is identified with L{p) and the parameter X is replaced 
by its characteristic value n. The polynomials obtained in the 
solution of the r equation for the hydrogen atom are hence the 
associated Laguerre polynomials of degree n — I — 1 and of 
order 21 + 1. Moreover, the wave functions in r are, except for 
normalizing factors, the functions 

These functions are called the associated Laguerre functions. 
We shall discuss them in detail in succeeding sections. 

It is easily shown from Equation 20-1 that the generating 
function for the associated Laguerre polynomials of order s is 1 

00 pU 

u.(p, «) - 2^r ur - ( " V' ii-uy^ '- (20_8) 

The polynomials can also be expressed explicitly: 

n-l-l 

L 2i +Hp) = ^ /_ n *+i i(n + l)\}> 

L n+l ( P ) j^k i; (n _ j _ ! _ fc)!(2 j + x + fc)!fcI P • 

fc=0 

(20-9) 

1 This was given by Schrodinger in his third paper, Ann. d. Phys. 80, 486 
(1926). 



132 THE HYDROGEN ATOM [V-21a 



In Appendix VII, it is shown that the normalization integral 
for the associated Laguerre function has the value 

Vp« {L#i>(p) } Vdp = ^iY- !)! ' (2 °- 10) 



1 



the factor p 2 arising from the volume element in polar coordinates. 
From this it follows that the normalized radial factor of the wave 
function for the hydrogen atom is 



with 



*"« = -V(S) 3 2:i(n + 0!}' e ^ P ' L " +V(p) ' (2 °- U) 

p - 2ar = **& - M,. (20 -12) 

r n/i 2 na 



Problem 20-2. Derive relations for the associated Laguerre polynomials 
and functions corresponding to those of Equations 20-2 and 20-3. 

21. THE WAVE FUNCTIONS FOR THE HYDROGEN ATOM 

21a. Hydrogenlike Wave Functions. — We have now found 
the wave functions for the discrete stationary states of a one- 
electron or hydrogenlike atom. They are 

+nim(r, tf, tp) = RnMGUmmM, (21-1) 



with 



and 



*mM = -)-«*-*, (21-2) 

^-{^^^w^r^ -^ (2i - 3) 



Rn,{r) = -[w &^ 



Z- 1)! 



+ 0M-J 

in which 



VVLJSiKp), (21-4) 



p = — r (21-5) 

na 



and 

«o = 



47T 2 /X6 2 



a being the quantity interpreted in the old quantum theory a3 
the radius of the smallest orbit in the hydrogen atom. The 



V-21a] THE WAVE FUNCTIONS FOR THE HYDROGEN ATOM 133 

functions P[ mI (cos #) are the associated Legendre functions 
discussed in Section 19, and the functions L^f^p) are the asso- 
ciated Laguerre polynomials of Section 20. The minus sign in 
Equation 21-4 is introduced for convenience to make the function 
positive for small values of r. 

The wave functions as written here are normalized, so that 

fo" fj fo"** 1 ^' *' *)*»'"( r ' *' ^ sin MrfMr = 1 - ( 21 ~ 6 ) 

Moreover, the functions in r, #, and <p are separately normalized 
to unity: 

£'*X<p)*„Md<p = 1, 
f r {Qim(#)} 2 sin #d# = 1,1 (21-7) 

f °° {R nl (r)} 2 r 2 dr = 1. 

They are also mutually orthogonal, the integral 

J["J[* f*"+nim(r, &, <p)+ n rAr, *, <f)r 2 sin *d<pd*dr 

vanishing except for n = n', I = l' } and m = m' '; inasmuch as if 
m 7* m\ the integral in <p vanishes; if m = m! ', but I ^ V ', the 
integral in # vanishes; and if m = m' and I = I' ', but n ^ n', 
the integral in r vanishes. 

Expressions for the normalized wave functions for all sets of 
quantum numbers out to n — 6, I = 5 are given in Tables 21-1, 
21-2, and 21-3. 

The functions $m(<p) are given in both the complex and the 
real form, either set being satisfactory. (For some purposes 
one is more convenient, for others the other.) 

Table 21-1. — The Functions & m M 

<t> (^) = — y~ Or $oO) = — J=^ 

V27T V 2ir 

$i(v?) = — p=c»* or $icos(^) = —^ cos <p 

1 1 

*-iW = — y=£r tip or *tain(*0 = —7= sin ip 
\/2w V* 



134 THE HYDROGEN ATOM [V-21a 

Table 21-1. — The Functions *m(¥>). — (.Continued) 

* 2 (^) - ~" -pzze™ Or *2coe(¥>) - — 7=-- cos 2<p 

-\Z2tt y/ic 

*_ 2 (*>) = — —e~ ivt> or **in(*0 = — 7^ sin 2<p 
y/2v -y/v 

Etc. 

Table 21-2.— The Wave Functions e* m (0) 
(The associated Legendre functions normalized to unity) 
I = 0, s orbitals: 

e o(t» = -y- 

/ » 1, p orbitals: 

©io(<>) = -r- cos t> 

V3 
e lafcl (0) = — - sin 



I = 2, d orbitals: 



a/10 

G 2O (0) - ~ — (3 cos 2 1? - 1) 
4 



/15 . 
62*1 (#) = • sin cos c q 

a/15 
e 2 -. 2 (0) = -—— sin 2 
4 

1 = 3,/ orbitals : 



, s 3Vl4/5 , \ 

e 8O (0) = v ( - cos 3 - cos 1 

-v/42 

B,*i(t?) = -^— sin 0(5 cos 2 0-1) 

o 

O,* 2 (0) = ~ — sin 2 cos 



a/70 
G 8 *8 W = ~~ sin 3 
8 

J = 4, gr orbitals: 

9a/2 /35 \ 

e 4O (0) = -~- ( ■— cos 4 0-10 cos 2 Hi] 

, N 9-\/l0 (l % \ 

9i*i W = — - — sin I - cos 3 — coft I 



V-21a] THE WA VE F UNCTIONS FOR THE H YDROGEN A TOM 135 

Table 21-2. — The Wave Functions Gj m (#).— (Continued) 
3\/5 

~ /on V gi n 2 ,y(7 CO g2 ^ _ 1) 



t>4* 


■2W 


= g on-, 


n< 


e 4 - 


->(*) 


3^70 . , 
8 


'* 


e 4 - 


.4(1?) 


3\/35 . A 


t> 



cost? 



I = 5, h orbitals: 

, x 15\/22/21 14 t \ 

©6o(#) = — ~ — I ™ cos 6 1? - -— cos 8 1? -f cos 1? ) 



16 \5 



6ui(« = 



'165 



16 



- sin #(21 cos 4 # - 14 cos 2 tf + 1) 



'1155 



8 
/770 



32 



sin 2 1?(3 cos 3 1> — cos #) 
sin 3 # (9 cos 2 - 1) 



x 3V385 . , 

0»*4W = — — — sin 4 & cos # 

16 

, N 3^154 . K 
e M W = 32 sin 6 * 



Table 21-3. — The Hydrogenlike Radial Wave Functions 

n = 1, K shell: 

— £ 
I = 0,1s ffio(r) = (Z/a )K'2e 2 

n = 2, L shell: 

(Z/o )» 



Z = 0, 2s ft, (r) = 

I = 1, 2p fl„(r) = 
n = 3, M shell: 

J = 0, 3s /2» (r) = 

J - 1, 3p «,i(r) - 



2\/2 

(Z/a )M 
2y/E 

(Z/oo)» 
9\/3 

(Z/ao)* 
9V6 



(2 - P )e 2 



-pe 



(6 - 6 P + P 2 )e 



(4 - P ) P e 2 



,^« 2 



J = 2, 3d * M (r) - ^£>>e~ 2 
9V30 



136 THE HYDROGEN ATOM [V-21a 

Table 21-3. — The Hydrogenlike Radial Wave Functions. — (Continued) 
n = 4, N shell: 

= 0, 4 8 R A0 (r) = (Z/ ^ o)/ (24 - 36p + 12 P 2 - p 3 )e~ 2 

= 1, 4p R Al (r) = --^L(20 - 10p + P *)pe 2 
32 VI 5 

= 2, 4d /2„(r) = {Z/ao) _ (6 - p^e" 2- 
96V5 

= 3,4/ ft.fr)- ^V* 

= 5, O^hell: 

(Z la )M -£ 

= 0, 5s R 60 (r) = - ' " (120 ~ 24 °p + 12 °p 2 ~ 2 °P 3 + P 4 )* 2 
300 V5 

(7 In )^ — - 

= 1, 5p « 6 i(r) = -^--==.(120 - 90 P + 18 P 2 - P 3 ) P e 2 
150 V 30 

/w/ yi^ __ p 

= 2, bd R M (r) = --^— °;1=(42 - 14 P + P 2 ) P 2 e 2 
150 V 70 

(7 In \¥* — - 

= 3, 5/ ft 63 (r) = -^-^-(8 - P ) P 3 e 2 
300 V 70 

= 4, 5gr R b i(r) = 7=P 4 e z 

900V70 

= 6, P shell: 

(Z/a ) H 
■ 0, 6s fleo(r) = -^-- -(720 - 1800 P + 1200p 2 - 300p 3 + 30 P 4 

2160V6 



6V 2 



-P 6 )e 

= 1, 6p R t i(jr) = 7= (840 - 840 P + 252 P 2 - 28 P 3 + P 4 ) P e 2 

432V 2 10 

= 2, 6d fl 62 (r) = °/-_ (336 - 168 P + 24 P 2 - P 3 ) P 2 e~ 2 

864V105 

= 3, 6/ R n (r) = (Z/a ° ) / L(72 - 18 P + P 2 )p 3 e~2 
2592 V 35 

= 4, 60 R*(r) = (Z/ao) ^ (10 - p^e"^ 
12960V7 

= 5, 6/i /?66(r) = 7==-P 6 e z 

12960 V77 



V-21a] THE WA VE FUNCTIONS FOR THE HYDROGEN ATOM 137 

The wave functions 9j m (#) given in Table 21-2 are the asso- 
ciated Legendre functions P| m| (cos #) normalized to unity. The 
functions P[ m ^ (cos &) as usually written and as defined by 
Equations 19-1 and 19-7 consist of the term sin lml # and the 
polynomial in cos # multiplied by the factor 

(* + M)i or q + M + p! 



Jl + \m\\Jl - \m\\ Jl + \m\ + lV/ (j - \m\ - l V 

as m + I is even or odd. Expressions for additional associated 
Legendre functions are given in many books, as, for example, 
by Byerly. 1 Numerical tables for the Legendre polynomials 
are given by Byerly and by Jahnke and Emde. 2 

Following Mulliken, we shall occasionally refer to one-electron 
orbital wave functions such as the hydrogenlike wave functions 
of this chapter as orbitals. In accordance with spectroscopic 
practice, we shall also use the symbols s, p y d, /, g, • • • to 
refer to states characterized by the values 0, 1, 2, 3, 4, • • • , 
respectively, of the azimuthal quantum number l y speaking, for 
example, of an s orbital to mean an orbital with 1 = 0. 

In the table of hydrogenlike radial wave functions the poly- 
nomial contained in parentheses represents for each function 
the associated Laguerre polynomial Ll^ip), as defined by 
Equations 20-1 and 20-5, except for the factor 

-(n + i)!/(n-i- 1)!, 

which has been combined with the normalizing factor and 
reduced to the simplest form. It is to be borne in mind that 
the variable p is related to r in different ways for different 
values of n. 

The complete wave functions 4<nim(r, #, <p) for the first three 
shells are given in Table 21-4. Here for convenience the variable 
p = 2Zr/na has been replaced by the new variable <r, such that 

n Z 

W. E. Byerly, "Fourier's Series and Spherical Harmonics," pp. 151, 
159, 198, Ginnand Company, Boston, 1893. 

2 W. E. Byerly, ibid., pp. 278-281; Jahnke and Emde, "Funktionen- 
tafeln," B. G. Teubner, Leipzig, 1933. 



138 THE HYDROGEN ATOM [V-21a 

The relation between <r and r is the same for all values of th° 
quantum numbers. The real form of the <p functions is used. 
The symbols p x , p v , p x , d x+v , d y + g , d x + x , d xyi and d z are introduced 
for convenience. It is easily shown that the functions ^ np ,, 
\l/ nPy , and \l/ nPa are identical except for orientation in space, the 
three being equivalently related to the x, y, and z axes, respec- 
tively. Similarly the four functions tfw, +1 ,, ^nd v+tf tnd x+t , and 
$nd tv are identical except for orientation. The fifth d function 
\l/ ndt is different. 

Table 21-4. — Hydrogenlike Wave Functions 
K Shell 
n = 1, I = 0, w = 0: 



= — (-Y -* 
\Ztt \ a oJ 

//Shell 



n = 2, I = 0, m = 0: 



1 /Z\H -- 

+u = 7=(-) (2-<r)6 2 

4 V2*- \ a o/ 

0: 

*i p- = 7= ( - ) ° e cos i> 

4V2*- \«o/ 

= ±1: 

i /zV* 

^2 Pj = 7= I — J <r 

4 V 2tt \Oo/ 

l /zV* -s . 
^ 2p = = ( — J ae 2 sin i? sin v 

4V2*- \ a °/ 



n = 2, J = 1, m = 0: 

n = 2, / = 1, m = ±1: 

i /zV __^ 

o-e 2 sin # cos <p 



M Shell 
n = 3, / = 0, m = 0: 

n - 3, / = 1, m = 0: 



i /zV -- 

*i. = 7= f - ) (27 - 18<r + 2<r 2 )e 3 

~ /3t V° / 



81 V^ 
n = 3, J = 1, m = ±1: 



V2 
^sp, = 7= ( — ) (6 — a)ae 3 COS t? 



(6 - <r)ae 3 I 



V?" /Z\ 

^i p , = 7= ( — 1 (6 — <r)<re 6 sin t> cos ^ 



V-21b] THE WAVE FUNCTIONS FOR THE HYDROGEN ATOM 139 

Table 21-4. — Hydrogenlike Wave Functions. — (Continued) 

^ / Z \H -£ 

tzp y = 7=-[ — ) (6 — <r)ae 6 sin # sin <p 

81 VV \«o/ 



n =: 3 ? / = 2, m = 0: 

81 V^ V ( 



^is x+t = -p I — ] (r 2 e 3 sin #cos t? cos v» 

^sd v+# = 7= I — | * 2 e sin # cos # sin <p 



1 /ZV* -- 

*, d , = -=, [ - ) <r 2 e 3 (3 cos 2 tf - 1) 

81V6ir \°o/ 

n = 3, J = 2, w = ± 1 : 

— [ — 1 <r 2 e * sin # cos t? 
81 Vt \ a °/ 

8lV^\ a °/ 
n = 3, J = 2. m = ±2: 

1 /ZV* - 5 

^3d Itf = 7= | — I <r l e 3 sin 2 # cos 2<p 

81V2x \ a V 

i /zV* -5 

^sd, +tf = 7=r [ — | <r 2 e 3 sin 2 1? sin %p 

81 V2ir \ fl o/ 

Z 

with <r = — r. 
a 

21b. The Normal State of the Hydrogen Atom. — The proper- 
ties of the hydrogen atom in its normal state (Is, with n = 1, 
I = o, m = 0) are determined by the wave function 

1 — 
^ 100 = : — - e ao . 



'jra: 



The physical interpretation postulated for the wave function 

1 -^ 
requires that \^*^ = — ge °° be a probability distribution function 

TTCLq 

for the electron relative to the nucleus. Since this expression 
is independent of & and <p f the normal hydrogen atom is spheri- 
cally symmetrical. The chance that the electron be in the 

1 -£ 
volume element r 2 dr sin &d&d(p is — z e °*r 2 dr sin &d$d<p, which 

TTCLq 

is seen to be independent of # and <p for a given size of the volume 
element. This spherical symmetry is a property not possessed 
by the normal Bohr atom, for the Bohr orbit was restricted to a 
single plane. 



140 



THE HYDROGEN ATOM 



[V-21b 



By integrating over # and <p (over the surface of a sphere), 
we obtain the expression 



4 -^ 
D(r)dr = ~^r 2 e ao dr 

cl 



as the probability that the electron lie between the distances 
r and r + dr from the nucleus. The radial distribution function 



2r 



100 



D\Qo(r) = —jr 2 e ao is shown in Figure 21-1 (together with ^ 
a 

and ^ioo) as a function of r, the distance from the nucleus. It 



* i 


\ 


.-., 


k 


4jrrWf 


f 1 ^ 




1 1 1 1 1 1 


( 


) 1.0 2.0 3.0% 



Fig. 21-1. — The functions \f/, \p*\p, and 4x7- VV for the normal hydrogen 
atom. The dashed curve represents the probability distribution function for a 
Bohr orbit. 



is seen that the probability that the electron remain within about 
1 A of the nucleus is large ; that is, the "size" of the hydrogen atom 
is about the same as given by the Bohr theory. Indeed, there is 
a close relation; the most probable distance of the electron from 
the nucleus, which is the value of r at which D(r) has its maximum 
value, is seen from Figure 21-1 to be a = 0.529A, which is just 
the radius of the normal Bohr orbit for hydrogen. 



V-21b] THE WA VE FUNCTIONS FOR THE HYDROGEN ATOM 141 

The distribution function itself is not at all similar to that 
for a circular Bohr orbit of radius a , which would be zero every- 
where except at the point r = a . The function ^ 00 has its 
maximum value at r = 0, showing that the most probable 
position for the electron is in the immediate neighborhood of 
the nucleus; that is, the chance that the electron lie in a small 
volume element very near the nucleus is larger than the chance 
that it lie in a volume element of the same size at a greater 
distance from the nucleus. 1 It may be pointed out that a Bohr 
orbit in the form of a degenerate line ellipse, obtained by giving 
the azimuthal quantum number k of the old quantum theory 
the value instead of the value 1, leads to a distribution function 
resembling the wave-mechanical one a little more closely. This 
is shown in Figure 21-1 by the dashed curve. The average 
distance of the electron from the nucleus, given by the equation 

fnlm = Ji ' ttnlnrtnl m r*dr SU1 M&d<p y (21~8) 

is found in this case to be equal to ^a . This is also the value 
calculated for the Bohr orbit with k — 0; in fact, it will be shown 
in the next section that for any stationary state of the hydrogen 
atom the average value of r as given by the quantum mechanics 
is the same as for the Bohr orbit with the same value of n and 
with k 2 equal to 1(1 + 1). It will also be shown in Chapter XV 
that the normal hydrogen atom has no orbital angular momen- 
tum. This corresponds to a Bohr orbit with fc = but not with 
k = 1. The root-mean-square linear momentum of the electron 
is shown in the next section to have the value 2irixe 2 /h y which is 
the same as for the Bohr orbit. We may accordingly form a 
rough picture of the normal hydrogen atom as consisting of an 
electron moving about a nucleus in somewhat the way cor- 
responding to the Bohr orbit with n = 1, k = 0, the motion 
being essentially radial (with no angular momentum), the 
amplitude of the motion being sufficiently variable to give rise 
to a radial distribution function D(r) extending to infinity, 
though falling off rapidly with increasing r outside of a radius 
of 1 or 2A, the speed of the electron being about the same as in 
the lowest Bohr orbit, and the orientation of the orbit being 

1 The difference between the statement of the preceding paragraph and 
this statement is the result of the increase in size of the volume element 
4rr 2 dr for the former case with increasing r. 



142 



THE HYDROGEN ATOM 



[V-21c 



sufficiently variable to make the atom spherically symmetrical. 
Great significance should not be attached to such a description. 
We shall, however, make continued use of the comparison of 
wave-mechanical calculations for the hydrogen atom with 
the corresponding calculations for Bohr orbits for the sake of 
convenience. 



5 


t 




4 


-\ 




3 


A 




2 


i \ 




1 


\ \^^ 


n-1, 1-0 






I 


\\ 


n-2, 1-0 


-? 


\ \ 




A x ~ 






\ 


n-3, 1-0 




1 



\^ 


n-2,l-l 






1 






n-3, 1-1 


. L 1 1 


1.. - J 1 L L I. - 



8 10 12 14 16 18 20 
?•+■ 

Fig. 21-2. — Hydrogen-atom radial wave functions R n i(r) for n = 1, 2, and 3 and 

I = and 1. 

21c. Discussion of the Hydrogenlike Radial Wave Functions. 

The radial wave functions R n i(r) for n = 1, 2, and 3 and I = 
and 1 are shown plotted in Figure 21-2. The abscissas represent 
values of p; hence the horizontal scale should be increased by the 
factor n in order to show R (r) as functions of the electron-nucleus 
distance r. It will be noticed that only for s states (with I = 0) 
is the wave function different from zero at r = 0. The wave 
function crosses the p axis n — I — 1 times in the region between 
p = and p = oo . 



V-21c] THE WA VE FUNCTIONS FOR THE HYDROGEN A TOM 143 

The radial distribution function 

Dni(r)=rMftni(0} 2 (21-9) 

is represented as a function of p for the same states in Figure 
21-3. It is seen from Figures 21-2 and 21-3 that the probability 
distribution function ^V> which is spherically symmetrical 
for s states, falls off for these states from a maximum value at 
r = 0. We might say that over a period of time the electron 



0-8 



0-4 




08 




I 0-4 

E o 

c 

Gc 

t^ 0-4 

^ 

0-8 

0-4 



0-4 





n-1, 1-0 





10 
9" 



12 14 16 18 20 



Fig. 21-3. — Electron distribution functions 4xr t [R n i(r)] 1 for the hydrogen atom. 

may be considered in a hydrogen atom in the normal state to 
form a ball about the nucleus, in the 2s state to form a ball and 
an outer shell, in the 3s state to form a ball and two concentric 
shells, etc. The region within which the radial distribution 
function differs largely from zero is included between the values 
of r at perihelion and aphelion for the Bohr orbit with the same 
value of n and with k 2 = 1(1 + 1), as is shown by the heavy 
horizontal line for each curve in Figure 21-3, drawn between the 
minimum and maximum values of the electron-nucleus distance 



144 THE HYDROGEN ATOM [V-21c 

for this Bohr orbit in each case. For these s orbits (with k = 0) 
the heavy line extends to r = 0, corresponding to a line ellipse 
with vanishingly small minor axis, in agreement with the large 
value of yf/*}// at r = 0. For states with I > 0, on the other hand, 
\f/*\l/ vanishes at r = 0, and similarly the minimum value of r 
for the Bohr orbits with k = \/l(l + 1) is greater than zero. 

The average distance of the electron from the nucleus, as given 
by Equation 21-8, is found on evaluating the integral to be 



Tnlm — 7* 



M{-'-^}] <«-»» 



The corresponding values of p are represented by vertical lines in 
Figure 21-3. From this expression it is seen that the size of the 
atom increases about as the square of the principal quantum 
number n, f n im being in fact proportional to n 2 for the states 
with 1 = and showing only small deviations from this propor- 
tionality for other states. This variation of size of orbit with 
quantum number is similar to that of the old quantum theory, 
the time-average electron-nucleus distance for a Bohr orbit 
being 



rrao 



{ i +&-$)}■ < 2i - u > 



which becomes identical with the wave-mechanical expression 
if k 2 is replaced by 1(1 + 1), as we have assumed in the foregoing 
discussion. 

Formulas for average values of various powers of r are given 
below. 1 It is seen that the wave-mechanical expressions as a 
rule differ somewhat from those of the old quantum theory, 
even when k 2 is replaced by 1(1 + 1). 

Average Values* of r 9 
Wave Mechanics 



-ft'+if 1 -*^}] 



* Expressions for r are given in Equations 21-10 and 21-11. 



1 1. Waller, Z. f. Phys. 38, 635 (1926); expressions for 
are given by J. H. Van Vleck, Proc. Roy. Soc. A 143, 679 (1934). 



(?)-(*) 



V-21c] THE WAVE FUNCTIONS FOR THE HYDROGEN A TOM 145 



Average Values of r*. — (Continued) 

Z 

a Q n 2 



(A 

\-V aWd + H 



Z 3 



3 W + HW + 1) 



2 ( 3n 2 j 



Old Quantum Theory 



¥{■+£-£)} 



Z 

aon 2 



M- — 

\r 3 / ajn 3 /c 3 
\r 4 / a,!n 3 /c 6 



To illustrate the use of these formulas, let us calculate the 
average potential energy of the electron in the field of the 
nucleus. It is 

Vmm =-111 +: im ~+«i m r>dr sin #d#d<p 






= -Ze 
Z 2 e 2 



(21-12) 



Now the total energy W , which is the sum of the average ki- 
netic energy f and the average potential energy V, is equal to 
— Z 2 e 2 /2aon 2 . Hence we have shown that the total energy is 
just one-half of the average potential energy, and that the average 



146 THE HYDROGEN ATOM [V-21d 

kinetic energy is equal to the total energy with the sign changed, 
i.e., 

'•'- - !£■ (21 - 13) 

This relation connecting the average potential energy, the 
average kinetic energy, and the total energy for a system of 
particles with Coulomb interaction holds also in classical mechan- 
ics, being there known as the virial theorem (Sec. 7a). 
Now we may represent the kinetic energy as 

T = ±(vl + Vl + Vl), 

in which p X) p V) and p z represent components of linear momentum 
of the electron and nucleus relative to the center of mass (that 
is, the components of linear momentum of the electron alone 
if the small motion of the nucleus be neglected). Hence the 
average value of the square of the total linear momentum 
P 2 = vl + 7>l + Vl is equal to 2/x times the average value of the 
kinetic energy, which is itself given by Equation 21-13 for both 
wave mechanics and old quantum theory. We thus obtain 

**» = "2o^ = \-iar) (21 " 14) 

as the equation representing the average squared linear momen- 
tum for a hydrogenlike atom in the wave mechanics as well as in 
the old quantum theory. This corresponds to a root-mean- 
square speed of the electron of 

/= 2*Ze 2 



nh 



(21-15) 



which for the normal hydrogen atom has the value 2.185 X 10 8 
cm/sec. 

Problem 21-1. Using recursion formulas similar to Equation 20-2 (or 
in some other way) derive the expression for f«j m . 

21d, Discussion of the Dependence of the Wave Functions on 
the Angles d and $. — In discussing the angular dependence of 
hydrogenlike wave functions, we shall first choose the complex 
form of the functions <£>(^>) rather than the real form. It will be 
shown in Chapter XV that there is a close analogy between the 



V-21d] THE WA VE FUNCTIONS FOR THE HYDROGEN A TOM 147 

stationary states represented by these wave functions and the 

Bohr orbits of the old quantum theory in regard to the orbital 

angular momentum of the electron about the nucleus. The 

square of the total angular momentum for a given value of I 

h 2 
is 1(1 + 1)t-2> an( * the component of angular momentum along 

the z axis is mh/2Tr, whereas the corresponding values for a Bohr 
orbit with quantum numbers nkm are k 2 h 2 /4w 2 and mh/2w, 
respectively. We interpret the wave functions with a given 
value of I and different values of m as representing states in which 
the total angular momentum is the same, but with different 
orientations in space. 

It can be shown by a simple extension of the wave equation 
to include electromagnetic phenomena (a subject which will 
not be discussed in this book) that the magnetic moment asso- 
ciated with the orbital motion of an electron is obtained from the 
orbital angular momentum by multiplication by the factor 
e/2m c, just as in the classical and old quantum theory (Sec. Id). 
The component of orbital magnetic moment along the z axis is 

hence mj- ; and the energy of magnetic interaction of this 

moment with a magnetic field of strength H parallel to the z axis 

he rj 

is m-. H. 

4rt"moC 

In the old quantum theory this spatial quantization was sup- 
posed to determine the plane of the orbit relative to the fixed 
direction of the z axis, the plane being normal to the z axis for 
m = ±k and inclined at various angles for other values of m. 
We may interpret the probability distribution function ^*^ in a 
similar manner. For example, in the states with m = ±1 
the component of angular momentum along the z axis, mh/2w, 
is nearly equal to the total angular momentum, \/l(l + l)h/2w, 
so that, by analogy with the Bohr orbit whose plane would be 
nearly normal to the z axis, we expect the probability distribution 
function to be large at & = 90° and small at & = 0° and 180°. 
This is found to be the case, as is shown in Figure 21-4, in which 
there is represented the function {Qim(&)} 2 for m = ±1 and for 
I = 0, 1, 2, 3, 4, and 5. It is seen that as I increases the prob- 
ability distribution function becomes more and more concen- 
trated about the xy plane. 



148 



THE HYDROGEN ATOM 



[V-21d 




Fig. 21-4.— Polar graphs of the function [Oi m W] 2 for m = ± I and I = 0, 
1, 2, 3, 4, and 5, showing the concentration of the function about the xy plana 
with increasing I. 



V-21d] THE WA VE FUNCTIONS FOR THE HYDROGEN A TOM 149 

The behavior of the distribution function for other values of m 
is similarly shown in Figure 21-5, representing the same function 
f or I = 3 and m = 0, ± 1, ±2, ±3. It is seen that the function 
tends to be concentrated in directions corresponding to the 
plane of the oriented Bohr orbit (this plane being determined 
only to the extent that its angle with the z axis is fixed). 

With the complex form of the <p functions, these figures 
represent completely the angular dependence of the probability 




Fia. 21-5.- 



-Polar graphs of the function [6{ OT (t?)] 2 for I 
±2, and ±3. 



3 and m = 0, ±1. 



distribution function, which is independent of <p. The alterna- 
tive sine and cosine functions of <p correspond to probability 
distribution functions dependent on <p in the way corresponding 
to the functions sin 2 m<p and cos 2 nap. The angular dependence 
of the probability distribution function for s and p orbitals in 
the real form (as given in Table 21-4) is illustrated in Figure 21-6. 
It is seen that, as mentioned before, the function s is spherically 
symmetric, and the functions p x , p V) and p s are equivalent except 
for orientation. The conditions determining the choice of wave 
functions representing degenerate states of a system will be 
discussed in the following chapter. 



150 



THE HYDROGEN ATOM 



[V-21d 



A useful theorem, due to Unsold, 1 states that the sum of the 
probability distribution functions for a given value of I and all 
values of m is a constant; that is, 



+i 



X Oim(#)*m*(<p)e lm (&)* m (<p) = constant. (21-16) 




Px Py 

Fig. 21-6. — Polar representation of the absolute values of the angular wave 
functions for a and p orbitals. The squares of these are the probability distribu- 
tion functions. 

The significance of this will be discussed in the chapter dealing 
with many-electron atoms (Chap. IX). 

Problem 21-2. Prove Uns61d's theorem (Eq. 21-16). 
1 A. Unsold, Ann. d. Phys. 82, 355 (1927). 



CHAPTER VI 
PERTURBATION THEORY 1 

In case that the wave equation for a system of interest can be 
treated by the methods described in the preceding chapters, or 
can be rigorously treated by any amplification of these methods, 
a complete wave mechanical discussion of the system can be 
given. Very often, however, such a procedure cannot be carried 
out, the wave equation being of such a nature as to resist accurate 
solution. Thus even the simplest many-electron systems, the 
helium atom and the hydrogen molecule, lead to wave equations 
which have not been rigorously solved. In order to permit 
the discussion of these systems, which more often than not are 
those involved in a physical or especially a chemical problem, 
various methods of approximate solution of the wave equation 
have been devised, leading to the more or less accurate approxi- 
mate evaluation of energy values and wave functions. Of these 
methods the first and in many respects the most interesting is 
the beautiful and simple wave-mechanical perturbation theory, 
developed by Schrodinger in his third paper in the spring of 1926. 
It is especially fortunate that this theory is very much easier 
to handle than the perturbation theory which is necessary for 
the treatment of general problems in classical dynamics. 

Before we can discuss this method, however, we need certain 
mathematical results concerning the possibility of expanding 
arbitrary functions in infinite series of normalized orthogonal 
functions. These results, which are of great generality and 
widespread utility, we shall discuss in the next section without 
attempting any complete proof. 

22. EXPANSIONS IN SERIES OF ORTHOGONAL FUNCTIONS 

The use of power series to represent certain types of functions 
is discussed in elementary courses in mathematics, and the 
theorems which state under what conditions the infinite series 

1 A generalized perturbation theory will be discussed in Section 27a. 

151 



152 PERTURBATION THEORY [VI-22 

obtained by formal methods converge to the functions they are 
meant to represent are also well known. An almost equally 
useful type of infinite series, which we shall use very frequently, 
is a series the terms of which are members of a set of normalized 
orthogonal functions each multiplied by a constant coefficient. 
If fo(x) t fi(x), fi{x) y • • • are members of such a set of normal- 
ized orthogonal functions, we might write as the series 

(p(x) = aafo(x) + aifi(x) + ^2/2(2) + • • * 

= Xanf n (z). (22-1) 

n=0 

If the series converges and has a definite sum <p(x), we may express 
Equation 22-1 by saying that the infinite series on the right of 
the equation represents the function <p(x) in a certain region of 
values of x. We may ask if it is possible to find the coefficients 
a n for the series which represents any given function <p(x). A 
very simple formal answer may be given to this question. If 
we multiply both sides of Equation 22-1 by ft(x) and then 
integrate, assuming that the series is properly convergent so 
that the term-by-term integration of the series is justified, 
then we obtain the result 

£*(x)ft{x)dx = Oft, (22-2) 

since 

£mx)f n (x)dx = if n * *, ) (22 _ 3) 

= 1 if n = k.) 
a ^ x ^ b defines the orthogonality interval for the functions 

/-(*)■ 

In many cases the assumptions involved in carrying out 
this formal process are not justified, since the series obtained may 
either not converge at all or converge to a function other than 
v?(x). Mathematicians have studied in great detail the condi- 
tions under which such series converge and have proved 
theorems which enable one to make a decision in all ordinary 
cases. For our purposes, however, we need only know that such 
theorems exist and may be used to justify all the expansions 
which occur in this and later chapters. 



VI-22] EXPANSIONS IN SERIES OF ORTHOGONAL FUNCTIONS 153 

The familiar Fourier series is only one special form of an 
expansion in terms of orthogonal functions. Figure 22-1, 
which gives a plot of the function 

<p(x) = 1 for < x < t, ) (22r-A) 

<p(x) = -1 for t < x < 2tt,/ 

together with the first, third, and fifth approximations of its 
Fourier-series expansion 

<p(x) = a + «i sin x + bi cos x + a 2 sin 2x + 

62 cos 2x + • • • , (22-5) 

illustrates that a series of orthogonal functions may represent 
even a discontinuous function except at the point of discontinuity. 




Fig. 22-1. — The function <p{x) = + 1 f or < x < w, -1 for tt < x < 2ir, and 
the first, third, and fifth Fourier-series approximations to it, involving terms to 
oin x, sin 3x, and sin 5x, respectively. 

If we had evaluated more and more terms of Equation 22-5, 
the series would have approached more and more closely to the 
function <p(x), except in the neighborhood of the discontinuity. 

The most useful sets of orthogonal functions for our purposes 
are the wave functions belonging to a given wave equation. In 
preceding chapters we have shown that the solutions of certain 
wave equations form sets of normalized orthogonal functions, 
such as for example the Hermite orthogonal functions which 



154 PERTURBATION THEORY [VI-22 

are the solutions of the harmonic oscillator problem (Sec. 11). 
In Appendix III it is shown that the solutions of any wave 
equation form such a set of orthogonal functions. 

In making expansions in terms of orthogonal functions, it is 
necessary to be sure that the set of junctions is cam,pLeLe. Thus in 
the example of Equations 22-4 and 22-5, if we had used the set 
cos x, cos 2x, • • • , without the sine terms, the series obtained 
would have converged, but not to the function <p(x), because the 
set of functions cos x, cos 2x, • • • is not complete. This 
requirement of completeness necessitates that all the solutions 
of the wave equation be included when using these solutions for 
an expansion of an arbitrary function. Since many wave equa- 
tions lead to a continuous spectrum of energy levels as well as a 
discrete spectrum, it is necessa ry to incl ude the wave f unction^ 
belonging 4o the continuous levels when making an expansion. 
The quantum numbers for the continuous spectra do not have 
discrete values but may vary continuously, so that the part 
of the expansion involving these wave functions becomes an 
integral instead of a sum as in Equation 22-1. 

However, in many special cases it is easy to see that certain 
of the coefficients a* will be zero so that in those cases an expan- 
sion is possible in a set of functions which is not complete. Thus 
if the function <p(x) which we are attempting to represent is an 
even function 1 of x y and if the orthogonal set we are using for the 
expansion contains both even and odd functions, the coefficients 
of all the odd functions fk{x) will vanish, as may be seen from the 
consideration of Equation 22-2. 

All the ideas which have been discussed in this section can be 
generalized without difficulty to systems of several variables. 
Normalized orthogonal functions in several variables Xi, yi, 
• • • , z N satisfy the condition * 

/ " • " !fi(*u Vu ' " ' f **)/m(xi, 2/i, • • • , z N )dr 

-Oifn*m,i (22 _ 6) 
= 1 if n = m, ) 

in which the integration is carried out over the whole of the 
configuration space for the system, and dr is the volume element 

1 The function f(x) is called an even function of x if f(—z) is equal to 
f(x) for all values of x, and an odd function of x if /( — x) is equal to ~/(a?) 
for all values of x. 



VI-22] EXPANSIONS IN SERIES OF ORTHOGONAL FUNCTIONS 155 

for the particular coordinate system in which the integral is 
expressed. Orthogonal functions in several variables usually 
are distinguished by several indices, which may however be 
symbolized by a single letter. An example of a three-dimensional 
set of normalized orthogonal functions is the set of solutions of 
the wave equation for the hydrogen atom. We have obtained in 
Chapter V the solutions belonging to the discrete levels; the 
quantum numbers nlm provide the indices for these functions. 
The solutions for the continuous spectrum of the atom, i.e., the 
system resulting when the electron has been completely removed 
from the nucleus, must be included if a complete set is desired. 1 
The coefficients in the expansion of an arbitrary function of 
several variables are obtained from an equation entirely analogous 
to Equation 22-2, 

a* = / • • • JVtei, Vu * * ' , z*)/*(zi, Vu ' ' ' , Zn)<It } (22-7) 

in which the limits of integration and the meaning of dr are tne 
same as in Equation 22-6. 

A function <p which is expressed in terms of the normalized 
functions of a complete orthogonal set is itself normalized if the 

coefficients in the "expansion satisfy the relation 2ja*a n = 1. 

n 

It may be mentioned that in some cases it is convenient to 
make use of complete sets of functions which are not mutually 
orthogonal. An arbitrary function can be expanded in terms 
of the functions of such a set; the determination of the values 
of the coefficients is, however, not so simple as for orthogonal 
functions. An example of an expansion of this type occurs in 
Section 24. 

In certain applications of expansions in terms of orthogonal 
functions, we shall obtain expressions of the form 

Xa n fn(x) = 0. 

n 

By multiplying by /*(x) and integrating, we see that the coeffi 
cient of each term must be zero; i.e., a n = for all values of n. 

1 For a discussion of the wave functions for the continuous spectrum of 
hydrogen, see Sommerfeld, "Wave Mechanics," p. 290. 



156 PERTURBATION THEORY [VI-23 

Problem 22-1. Obtain the first four coefficients in the expansion <p(x) = 
(2T + *)-H = ^ ak yJ^-^P k (x) f where P k (x) is the fcth Legendre poly- 
nomial given in Section 19. This expansion is valid only for \x\ $ 1. Plot 
ip(x) and the approximations to it given by including the first, second, third, 
and fourth terms of the expansion. If possible, obtain a general expression 
for an, using the generating function for Pk(x). 

^3. FIRST-ORDER PERTURBATION THEORY FOR A 
NON-DEGENERATE LEVEL 

In discussing many problems which cannot be directly solved, a 
solution can be obtained of a wave equation which differs from 
the true one only in the omission of certain terms whose effect 
on the system is small. Perturbation theory provides a method 
of treating such problems, whereby the ap proximate eq uation 
js first solved and th gnjji e sm a^additional terms are introduced 
as correcti ons^ 

Let us write the true wave equation in the form 

m - W+ = 0, (23-1) 

in which H represents the operator 



H = 8. 



?2i* + F - (23 - 2) 



We assume that it is possible to expand H in terms of some 
parameter X, yielding the expression 

H = H° + \H' + X 2 #" + • • ' 1 (23-3) 

in which X has been chosen in such a way that the equation to 
which 23-1 reduces when X — > 0, 

#<y,o _ W o r = 0> (23_4) 

can be directly solved. This equation is said to be the wave 
equation for the unperturbed system, while the terms 

X#' + X 2 #" + • • • 

are called the perturbation. As an illustration, we might men- 
tion the problem of the Stark effect in atomic hydrogen, in 
which an electric field is applied to the atom. In this problem 
the field strength E provides a convenient parameter in terms 



VI-23] FIRST-ORDER PERTURBATION THEORY 157 

of which the Hamiltonian may be expanded. When E is zero, 
the problem reduces to that of the ordinary hydrogen atom, 
which we have already solved. 
The unperturbed equation 23-4 has solutions 

+1, K K • • • , *2, • • • , 

called the unperturbed wave functions, and corresponding energy 
values 

wi m w% • • • , wi, - • • . 

The functions $! form a complete orthogonal set as discussed in 
Section 22, and, if we assume that they have also been normalized, 
they satisfy the equation (Appendix III) 



WW 



T = o if i * j\ 

= i if i = j.f 



(23-5) 



Now let us consider the effect of the perturbation. By hypoth- 
esis it will be small, and from the continuity properties of wave 
functions 1 we know that the energy values and wave functions 
for the perturbed system will lie near those for the unperturbed 
system. In other words, the application of a small perturbation 
is not going to cause large changes. With these facts in mind 
we can expand the energy W and the wave function \p for the 
perturbed problem in terms of X and have reasonable assurance 
that the expansions will converge, writing 

fa = V k + WL + XVr + • • • (23-6) 

and 

fa = Wl+ ^l + VW k ' + ' * ' . (23-7) 

If the perturbation is really a small one, the terms of these series 
will become rapidly smaller as we consider the coefficients of 
larger powers of X; i.e., the series will converge. 

We now substitute these expansions for H> fa, and W k into 
the wave equation 23-1, obtaining the result, after collecting 
coefficients of like powers of X, 

(H«V k - Wl+l) + {H°+' k + H'+l - WM - WM)\ 

+ (H%' + H'i' k + H"W - WW - WM - W'M)\* 

+ • • • =0. (23-8) 

1 Discussed, for example, in Courant and Hilbert, " Methoden der mathe- 
matischen Physik." 



158 PERTURBATION THEORY [VI-23 

If this series is properly convergent, we know that in order for it to 
equal zero for all values of X the coefficients of the powers of X 
must vanish separately. 1 fyhe coefficient of X° when equated to 
zero gives Equation 23-4, ip that we were justified in beginning 
the expansions 23-6 and 23^7 with the terms \p° and W°. The 
coefficient of X gives the equation 

H°+' k - Wfr' k = (W' k - ff')«. (23-9) 

To solve this we make use of the expansion theorem discussed 
in the last section. We consider that the unknown functions 
\f/ f k can be expanded in terms of the known functions ^?, since the 
latter form a normalized orthogonal set, and write 

ti = X a ^°' (23-10) 

i 

(The coefficients a t might be written as an, but we shall assume 
throughout that we are interested only in the state k and there- 
fore shall omit the second subscript.) Using this, we obtain the 
result 

i i 

since 

tf<ty? = WW. 

Equation 23-9 therefore assumes the form 

XatiW*, - WWf = (W' k - H'M. (23-12) 

If we multiply by ypl * and integrate over configuration space, we 
observe that the expression on the left vanishes : 

J*8*2ai(W7 - WMdr *> Joi(lF?- WDM **?&■ = 0, 

i I 

since JV£ *tfdr vanishes except for I = k t and for this value 

1 Thus, if 

^a n X» ss *>(x) s 0, 
n 

then, assuming that the series is properly convergent, we can write 



M^) - o. 

n!\d\ tt /x~o 



VI-23] FIRST-ORDER PERTURBATION THEORY 159 

of I the quantity W? — W k vanishes; and hence we obtain the 
equation 

ir(ir;-ff# = o. (23-13) 

This solves the problem of the determination of W k , the first- 
order correction to the energy. Since W k is a constant in 
Equation 23-13, the integration of the term containing it can be 
carried out at once, giving the result, when multiplied by X, 

W k =\M*HW k dr. (23-14) 

Since the correction to the energy is \W f k , it is convenient to 
include the parameter X in the symbols for the first-order pertur- 
bation and the first-order energy correction, so that to the 
first order it is usual to write the relations 

H = H° + H' y ) 

W k = W° k + W'J 
in which 

W' k = SH *H'+ldr. (23-16) 

This expression for the perturbation energy can be very simply 
described : The first-order -perturbation energy for a non-degenerate 
state of a system is just the perturbation function averaged over the 
corresponding unperturbed state of the system. 

We can also evaluate the correction \f/' k for the wave function. 
Multiplying each side of Equation 23-12 by tf *, we obtain, after 
integration^ 

a,(W? - Wl) = -/*? *#Wr, j ^ k, (23-17) 

where we have utilized the orthogonality and normalization 
properties of the ^ 0, s. The coefficients a ; in the expansion 23-10 
of yp' in terms of the set ip? are thus given by the relation 

■V-W^W' i-*- (23-18, 

The value of a k is not given by this process; it is to be chosen 
so as to normalize the resultant \p, and, if only first-order terms 
are considered (terms in X 2 neglected), it is equal to zero. It is 
convenient to introduce the symbol 

H' ik = M*H'fflT, (23-19) 



160 PERTURBATION THEORY [VI-23a 

so that the expression for the first-order wave function of the 
system, on introducing the above values of the coefficients a,, 
becomes 

00 

*» = *!- x 2 V?-Vg *°' (23 ~ 20) 

y-o 

in which the prime on the summation indicates the omission of 
the term with j = k. 

As mentioned before, it is customary to include X in the defini- 
tion of H' as indicated in Equation 23-15, so that we get finally 
for the first-order energy and the first-order wave function the 
expressions 

W k = Wl + H' kk (23-21) 

and 

oo 

+» = +1 - S' w7-V ' 9 - (23_22) 

23a. A Simple Example : The Perturbed Harmonic Oscillator. 

As a simple illustration of first-order perturbation theory we shall 
obtain the approximate energy levels of the system whose wave 
equation is 



dx* 



+ ^™(w - \kx* - ax* - bx*\ = 0. (23-23) 



We recognize that if a and b were zero this would be the wave 
equation for the harmonic oscillator, whose solutions we already 
know (Sec. 11). If a and b are small, therefore, we may treat 
these terms as perturbations, writing 

H' = ax* + bx\ (23-24) 

We need then to evaluate the integrals 

H '*n = a f^y°n * X W dx + 6 /.V ^ **WdX. (23"25) 

Since x 8 is an odd function and \pl *\pl an even function, the first 
of these integrals is zero, so that the first-order perturbation due 
to ay? is zero. To calculate the second integral we refer back to 



V-23a] FIRST-ORDER PERTURBATION THEORY 161 

Section lie for the functions ^° and their properties. Substi- 
tuting for xpl from Equation 11-20 we obtain the integral 

1 = j_\+»* x w dx = SX/" i2 ^ (t) ^^ (23_26) 

From Equation 11-15 we see that 

fff.({) = ^H» +1 (f) + tt#n-i(a (23-27) 

so that, after applying Equation 23-27 to %H n +x and f// n -i and 
collecting terms, we obtain the equation 

eHn(Z) = KHn+tiZ) + {n + K)ffn(«) + n( W - l)ffn- 2 (f). 

(23-28) 

By this application of the recursion formula for H n (£) we have 
expressed £ 2 H n (£) in terms of Hermite polynomials with constant 
coefficients. By squaring this we obtain an expression for 
£ A Hl(£), which enables us to express the integral in Equation 
23-26 as a sum of integrals of the form 

f~j-* t H n (t)H m (Z)dl; = if m * nl (23 _ 2 9) 

= 2 n n\yfic if m = n,) 

evaluated in Section lie. Thus we find for I the expression 

1 = ^^{iV" +2(n + 2)! + ( n + ii 2nni + 

w 2 (n - l) 2 2"- 2 (n - 2)\> 

when the value of N n given in Equation 11-21 is introduced. 
The first-order perturbation energy for this system is therefore 

W = H' M = ^- 2 (2n> + 2n + 1), 

so that the total energy becomes (to the first order) 

W = W° + W'=(n + ^jhp + ~(2n- + 2n + 1)^- 

(23-30) 



162 PERTURBATION THEORY [VT-23b 

In order to calculate the first-order wave function it would 
be necessary to evaluate all the quantities H ' nk . The x z term as 
well as the x 4 term will contribute to these integrals. The 
number of non-zero integrals is not, however, infinite in this 
case but quite small, only the terms with k = n, n ± 1, n ± 2, 
n ± 3, and n ± 4 being different from zero. 

23b. An Example: The Normal Helium Atom. — As another 
example of the application of first-order perturbation theory let 
us discuss the normal state of the helium atom. Since the term 
which we shall use as the perturbation is not particularly small, 
we must not expect an answer of very great accuracy. The 
potential energy for a system of two electrons and a nucleus of 
charge +Ze is 

Ze 2 Ze 2 p 2 
F= -— -t±+±, (23-31) 

ri r 2 r i2 

in which r x and r 2 are the distances of electrons 1 and 2, respec- 
tively, from the nucleus, and r J2 is the separation of the two 
electrons. If we make the approximation of considering the 
nucleus at rest, which introduces no appreciable error, the wave 
equation (see Equation 12-8) for the two electrons becomes 



H + = - 



&r 2 m 



/ ay ,^,^,^,!?V. <W\ 

\dx\ + dy\ + dz\ + dx\ + dy\ + dz\) 

( Ze 2 Ze 2 e 2 \ 
+ (- T-~ t-+^)+ = W+. (23-32) 



This equation applies to He, Li+, Be" 1 " 1 ", etc., with Z = 2, 3, 4, etc., 
respectively. The variables x u 2/1, Z\ are Cartesian coordinates 
of one electron, and x 2 , t/ 2 , z 2 those of the other; ra is the mass 
of the electron. 

Since if the term 6 2 /r i2 is omitted the wave equation which 
is obtained can be exactly solved, we choose this term as the 
perturbation function, 

p 2 
H' =— - 

The wave equation which remains, the unperturbed equation, 
can then be separated into two equations by the substitutions 

^°(zi, 2/1, *i, s 2 , 2/2, z 2 ) = u\(x h y h Zi)ul(x* } 2/ 2 , z 2 ) 



VI-23b] FIRST-ORDER PERTURBATION THEORY 163 

and 

w° = w\ + W° 2J 

the equation 1 for u\ being 



d 2 u\ dV x dhjj 8whn 0i 
dx\ + dy\ + dz\ + A* ' ' 



/^o + ^£_ 2 j w o = (23 _3 3) 



The equation for w§ is identical except for the changed subscripts. 
Equation 23-33 is just the hydrogenlike wave equation discussed 
in the preceding chapter, with solutions ^ n im{ri 9 #i> <Pi) and 
energy values — Z 2 W H /n 2 } in which 

27r 2 m 6 4 1Q KQ 
?/? = — p — = 13.53 v.e. 

The unperturbed wave function for the lowest level of the two- 
electron atom is therefore 

^100,100 = ^ioo(ri, #i, ^i)^ioo(r 2 , #2, <Pi) = 

u u (r h #1, <pi)u u (r 2 , #2, <p 2 ), (23-34) 

in which r\ } #1, <pi and r 2 , #2, ^2 are polar coordinates of the two 
electrons relative to axes with the nucleus at the origin. The 
corresponding energy value is 

TFSoo.100 = Wl + W\ = -2Z 2 W H . (23-35) 

The first-order perturbation energy W is the average value of 
the perturbation function H f = e 2 /r n over the unperturbed 
state of the system, with the value 



W = Jr*HWdr = j£*ioo.ioodr. (23- 



-36) 



From Table 21-4 of Chapter V we obtain for u u the expression 

tii. = *ioo = xffie"*, (23-37) 

\7ra 

in which p = 2Zr/a and <z = h 2 /±T 2 m e 2 . Using this in Equa- 
tion 23-34, we find for ^100,100 the expression 

^100,100 = — z e 2 e 2 . 

TTUq 

1 The symbol u will be used for the wave function for a single electron in a 
many-electron atom, with subscripts Is, 2s, 2p, etc. 



164 PERTURBATION THEORY [VI-23b 

The volume element is 

dr = r\dri sin &xd&id(pi • rfdr 2 sin # 2 d*M<p2, 
so that the integral for W becomes 
Ze 2 



W = 



9b » - — — Pi^Pi sin 0, 

* 7ra Jo Jo Jo Jo Jo Jo P12 

d$id<pipldp 2 sin # 2 d# 2 (i<p 2 , (23-38) 



in which p i2 = 2Zri 2 /a . 

The value of this integral is easily obtained, inasmuch as it 
corresponds to the electrostatic interaction energy of two spheri- 
cally symmetrical distributions of electricity, with density 
functions e~ pi and e~ P2 , respectively. In Appendix V it is shown 1 
that 

W' = */iZW H . (23-39) 

This treatment thus gives for the total energy the value 

W = -(2Z 2 - %Z)W«. (23-40) 

This may be compared with the experimental values of the total 
energy, which are obtained by adding the first and the second 
ionization energies. Table 23-1 contains, for He, Li + , Be ++ , 
B 3 +, and C 4_l_ , the experimental energy W exp ., the unperturbed 
energy W°, the total energy calculated by first-order perturbation 
theory W° + W, the difference A = W e x V . - W°, the difference 
A' = W^p. - W° - W, and finally the ratio -A'/A . 

It is seen that the error A' remains roughly constant in absolute 
value as the nuclear charge increases, which means that the 
percentage error decreases, since the total energy is larger for 
larger Z. This result is to be expected, inasmuch as for large 
nuclear charge the contribution of the attraction of the nucleus 
is relatively more important than that of the repulsion of the two 
electrons. It is pleasing that even in this problem, in which the 
perturbation function e 2 /r u is not small, the simple first-order 
perturbation treatment leads to a value of the total energy of 
the atom which is in error by only a small amount, varying from 
5 per cent for He to 0.4 per cent for C 4+ . 

1 This problem was first treated by A. Unsold, Ann. d. Phys. 82, 355 
(1927). 



VI-24] 



FIRST-ORDER PERTURBATION THEORY 



165 



Table 23-1. — Calculated and Observed Values of the Energy of 
Heliumlike Atoms and Ions 





-W txp .,v.e. 


-JP°, v.e. 


-W°-W', 
v.e. 


A , v.e. 


A', v.e. 


-A'/A° 


He 


78.62 


108.24 


74.42 


29.62 


-4.20 


0.142 


Li+ 


197.14 


243.54 


192.80 


46.40 


-4.34 


.094 


Be + + 


369.96 


432.96 


365.31 


63.00 


-4.65 


.074 


B 3+ 


596.4 


676.50 


591.94 


80.1 


-4.5 


.056 


C« + 


876.2 


974.16 


872.69 


98.0 


-3.5 


.036 



Problem 23-1. Calculate the first-order energy correction for a one- 
dimensional harmonic oscillator upon which the perturbation H'{x) acts, 
where H'(x) is zero unless |x| < e and H'(x) = b for \x\ < e, with e a quantity 
which is allowed to approach zero at the same time that b approaches infinity, 
in such a way that the product 2eb = c. Compare the effect on the odd and 
even levels of the oscillator. What would be the effect of a perturbation 
which had a very large value at some point outside the classically allowed 
range of the oscillator and a zero value elsewhere? 

Problem 23-2. The wave functions and energy levels of a particle in a 
one-dimensional box are given in Equations 14-6 and 14r-7. Calculate the 
first-order perturbation energy for such a system with a perturbation H' 
such that H' = b for (a/A;) - t ^ x ^ {a/k) -f e and H' = elsewhere, with 
€ — » as b — ► oo in such a way that 2eb = c, k being a given integer. With 
k = 5, determine which energy levels are the most and which are the least 
perturbed and explain. With k = 2, give the expression for the perturbed 
wave function, to the first order. 

Problem 23-3. Let H' be a perturbation, such that H'(x) = —b for 
^ x ^ a/2 and H'{x) ■= +b for a/2 ^ x ^ a, which is applied to a 
particle in a one-dimensional box (Eqs. 14-6 and 14-7). Obtain the first- 
order wave function. Show qualitatively that this function is such that the 
probability of finding the particle in the right-hand half of the box has been 
increased and explain in terms of classical theory. (Hint: Use the symmetry 
about the point x = a/2.) 



24. FIRST-ORDER PERTURBATION THEORY FOR A DEGENERATE 

LEVEL 

The methods which we have used in Section 23 to obtain the 
first-order perturbation energy are not applicable wjien the energy 
level of the unperturbed system is degenerate/for the reason 
that in carrying out the treatment we assumed that the perturbed 
wave function differs only slightly from one function ^ which 
is the solution of the unperturbed wave equation for a given 
energy value whereas now there are several such functions, all 



166 PERTURBATION THEORY [VI-24 

belonging to the same energy level, and we do not know which 
one (if any) approximates closely to the solution of the perturbed 
wave equation. 

An energy level Wk is called a-fold degenerate (see Sec. 14) 
when for W = Wk there exist a linearly independent wave func- 
tions \pk\f 4 / k2 f ^*3, • • • , ^ka satisfying the wave equation. 1 
Each of these is necessarily orthogonal to all wave functions for 
the system corresponding to other values of the energy (see 
Appendix III) but is not necessarily orthogonal to the other 
functions corresponding to the same value of the energy. Any 

a 

linear combination ]$►* Kj^/ k j of the wave functions of a degenerate set 

y-i 
such as ypki y ^*2, • • • , ^k<x is itself a solution of the wave equa- 
tion and is a satisfactory wave function corresponding to the 
energy IF*. We might therefore construct a such combinations 
Xk% by choosing sets of values for k, such that the different com- 
bin ations thus formed are linearly independent. "The^setrtjf 
functions so obtained, 

a 

Xki = 2) *f**/, t = 1, 2, 3, ■ ■ • , a, (24-1) 

is entirely equivalent to the original set \p k i, ^*2, • • • , ^*«. 
This indicates that there is nothing unique about any particular 
set of solutions for a degenerate level, since we can always con- 
struct an infinite number of other sets, such as x*i> • • • > Xka, 
which are equally good wave functions. The transformation 
expressed by Equation 24-1 is called a linear transformation 
with constant coefficients. 

It is usually convenient to deal with wave functions which 
are normalized to unity and which are mutually orthogonal 
Since the coefficients *»/ can always be chosen in such a way as 
to make the set x*» possess these properties, we shall ultimately 
assume that this has been done. 

Using these ideas, we can now investigate the application of 
perturbation theory to degenerate levels. We write the wave 
equation in the form 

1 The functions \pki, \frkt, • • • , tha are said to be linearly independent if 
there exists no relation of the form a^ki + aii>k% + • • • + Ocrtka = (in 
which a\, a t , • • • , a a are constant coefficients) which is satisfied for all 
rallies of the independent variables. 



VI-24] FIRST-ORDER PERTURBATION THEORY 167 

Hi - W* = (24-2) 

with 

H = H« + \H' + X 2 #" + • • • 

as before. The wave equation for the unperturbed system is 

#y> - W°V = 0, (24-3) 

the solutions of which are 

*Si, * o2, • • • ; *Si, *Si, •••;•••; *fc, *&, • • • , 

«-; • • • ; 

corresponding to the energy levels 

Wl W\; ■ ■ ■ ; Wli • • • • 

Now let us consider a particular wave function for the per- 
turbed equation 24-2. It is known, in consequence of the proper- 
ties of continuity of characteristic-value differential equations, 
that as the perturbation function \H' + • • • becomes smaller 
and smaller the energy value W of Equation 24-2 will approach 
an energy level of the unperturbed equation 24-3, Wl, say. 
The wave function under consideration will also approach more 
and more closely a wave function satisfying Equation 24-3. 
However, this limiting wave function need not be any one of 
the functions ^lu • • * > ^L> ft ma y be (and generally is) some 
linear combination of them. The first problem which must be 
solved in the treatment of a degenerate system is the determina- 
tion of the set of unperturbed wave functions to which the 
perturbed functions reduce when the perturbation vanishes; 
that is, the evaluation of the coefficients in the linear transforma- 
tion converting the initially chosen wave functions into the 
correct zeroth-order wave functions. These correct combinations, 
given by 

xli = X *»'*&'' * - 1, 2, • • • , a, (24-1) 

J'«i 

provide the first term of the expansion of \f/ki in powers of X, since 
by definition they are the functions to which the \puis reduce when 
X -> 0. Therefore 

*« = xli + Wki + XVii + • • ' (24-5) 

and 

Wki = Wt + \W' kl + W& + • • • , (24r^) 



168 PERTURBATION THEORY [VI-24 

where l(— 1, 2, • • • , a) designates the particular one of the a 
degenerate wave functions in question, are the equations which 
are analogous to Equations 23-6 and 23-7. (As in Equation 
23-10 we sometimes omit the subscript k; e.g., we write w 
for Kkiv) it must be borne in mind that throughout we are con- 
sidering the fcth degenerate level.) 

Substituting the expansions for \{/, W, and H into the wave 
equation 24-2, we obtain an equation entirely analogous to 
Equation 23-8 of the non-degenerate treatment, 

{H\l t - TF2x2i) + (HVi, + ff'xJi - ww kl - WM)X + 

• • • = 0, (24-7) 

from which, on equating the coefficient of X to zero as before, 
there results the equation (cf. Eq. 23-9) 

Wh - W° k r H = WUxlt - H'xl (24-8) 

So far our treatment differs from the previous discussion of 
non-degenerate levels only in the use of xu instead of \p ki ) i.e., 
in the introduction of a general expression for unperturbed 
functions instead of the arbitrary set \p ki . In the next step we 
likewise follow the previous treatment, in which the quantities 
\l/' k and H°\p k were expanded in terms of the complete set of 
orthogonal functions \f/ k . Here, however, we must in addition 
express x& in terms of the set \fr kl >, by means of Equation 24-4, 
in which the coefficients k iV are so far arbitrary. Therefore we 
introduce the expansions 

tii = XawrMv (24-0) 

kT 

and 

flV« = XawrHWr = %a kl k>i>WW k , v (24-10) 

h'V k'V 

into Equation 24-8 together with the expression for xli given by 
Equation 24-4. The result is 

a 

k'v r-i 

in which the right-hand side involves only functions \// kV belonging 
to the degenerate level Wl while the expansion on the left includes 



VI-24] FIRST-ORDER PERTURBATION THEORY 169 

all the $j*>i>'8. If we now multiply both sides of this equation by 
4*1* and integrate over configuration space, we obtain the result 

a 

MtH'Mvdr). (24-12) 

The left side of this equation is zero because \f/ ki and $j r are 
orthogonal if k ^ fc' and TTg. - F£ is zero if fc = fc'. If we 
introduce the symbols 

E' iV = MfH'^dr (24-13) 

and 

**v = R%°A, (24-14) 

we may express Equation 24-12 in the form 

2V(#Jr ~ A ? rF^) =0, j = 1, 2, 3, - • • , a. (24-15) 
r-i 

This is a system of a homogeneous linear simultaneous equations 
in the a unknown quantities kh, ki 2 , • • • , k&*. Written out in 
full, these equations are 

{H' u - &nW' kl )Kn + (H[ 2 - A lt WU)*it + ' ' ' + 

(H' la - A la TTi,)ic te = > ] 

(ff 21 - A 2 i^)k Z i + .(# 22 - A 22 J^)k I2 + • • • + 

(# 2a - AtoWMicta = 0,> (24-16) 

(H' al - AaiW' kl )Kn + (£Ti s - A a ,W' kl )K l2 + * * * + 

Such a set of equations can be solved only for the ratios of the 
k's; i.e., any one k may be chosen and all of the others expressed 
in terms of it. For an arbitrary value of W kU however, the set 
of equations may have no solution except the trivial one kw — 0. 
It is only for certain values of W' kl that the set of equations has 
non-trivial solutions; the condition that must be satisfied if 
such a set of homogeneous linear equations is to have non-zero 
solutions is that the determinant of the coefficients of the 
unknown quantities vanish ; that is, that 



170 



PERTURBATION THEORY 



[VI-24 



H' n - AulPS, H' u - A u W' kl 
H'n-AuWU H' n -A 2i W' kl 



H' la - A la W' kl 
H' ta - AtoW'u 



H' al -A al W' kl Hia-A^W'u 



"act &aaW k i 



= 0. 



(24-17) 

This determinantal equation can be expanded into an algebraic 
equation in W' hl which can then be solved for W ki . For the types 
of perturbation functions which arise in most physical and chemi- 
cal problems the determinant is either symmetrical about the 
principal diagonal, if the elements are real, or else has the property 
that corresponding elements on opposite sides of the principal 
diagonal are the complex conjugates of each other; that is, 
H' {i = H'f. In consequence of this property it can be shown 
that the determinant possesses a real roots, W' kl , W k2 , • • • , W' ka - 
These are the values of the first-order perturbation energy for the 
a wave functions which correspond to the a-fold degenerate 
unperturbed energy level W k . It may happen, however, that 
not all of the roots W kl , etc., are distinct, in which case the 
perturbation has not completely removed the degeneracy. 

The coefficients kh> which determine the correct zeroth-order 
wave function xli corresponding to any perturbed level W kt 
may be determined by substituting the value found for W kl 
into the set of simultaneous equations 24-16 and solving for the 
other coefficients in terms of some one of them. This remaining 
arbitrary coefficient may be adjusted so as to normalize xli- 
This process does not give uriique results if two or more roots W k i 
coincide, corresponding to the fact that since there still remains 
a certain amount of degeneracy the wave functions for the 
degenerate level are not uniquely determined but are to a 
certain degree arbitrary. 

If the original wave functions fin ' ' ' t W« were normalized 
and mutually orthogonal (which we have not hitherto needed 
to assume), the function Ayr is unity for j = V and zero other- 
wise, so that the determinantal equation 24-17 assumes the 
form 



#n — W k 'i H'n H[ z 

H' 21 H' 2% — W' M #« 



H ai 



H** 



Hal 



H ia 
H 2 a 



HL - W' kl 



= 0. 



(24-18) 



VI-24] FIRST-ORDER PERTURBATION THEORY 171 

An equation such as 24-17 or 24-18 is often called a secular 
equation, and a perturbation of the type requiring the solution 
of such an equation a secular perturbation. 1 

It is interesting to note that in case the secular equation has 
the form 



H' n - W' kl 

H 22 -W kl 



H'aa-WL 



0, 

(24-19) 



then the initially assumed functions yp kx , \l/ k2 , • • • > Vka are the 
correct zeroth-order functions for the perturbation H', as is seen 
on evaluation of the coefficients k of Equations 24-16. A secular 
equation in which all the elements are zero except along the 
principal diagonal is said to be in diagonal form. The roots 
W' kl are of course immediately obtainable from an equation in 
this form, since the algebraic equation equivalent to it is 

(ffii " W' kl )(H' 22 - W' kl ) - - • (HL - W' kl ) = 0, (24-20) 

with the roots W' kl = H' u , H 22y • • • , H' aa . 

1 In this sense secular means "accomplished in a long period of time" 
(Latin saecidum = generation, age). The term secular perturbation was 
introduced in classical mechanics to describe a perturbation which produces 
a slow, cumulative effect on the orbit. If a system of sun and planet, for 
which the unperturbed orbits are ellipses of fixed size, shape, and orientation, 
were perturbed in such a way as to change the law of force slightly from the 
inverse square, as is done, for example, by the relativistic change of mass 
with change of speed, the position of the major axis in space would change 
by a small amount with each revolution of the planet, and the orbit would 
carry out a slow precession in its own plane, with a period which would be 
very long if the magnitude of the perturbation were small. Such a perturba- 
tion of the orbit is called a secular perturbation. 

On the other hand we might have a system composed of a wheel in a 
gravitational field rotating about a horizontal frictionless axle passing 
through its center of mass and perturbed by the addition of a small weight 
at some point on its periphery in such a way as to accelerate the motion 
as the weight moves down and to decelerate it as the weight moves up. 
Such a perturbation, which produces a small effect on the motion with the 
high frequency characteristic of the original unperturbed motion of the 
system, is not a secular perturbation. 

The significance of the use of the word secular in quantum mechanics 
will be seen after the study of the perturbation theory involving the time 
(given in Chap. XI). 



172 



PERTURBATION THEORY 



[VT-24a 



This equation 24-19 illustrates, in addition, that the integrals 
H' mn depend on the set of zeroth-order functions ^li which is 
used to define them. Very often it is possible to guess in advance 
which set of degenerate ^ 0, s to use for a given perturbation in 
order to obtain the simplest secular equation. In particular, 
in case that the perturbation is a function of one variable (x, say) 
alone, and each function of the initial set of unperturbed wave 
functions can be expressed as the product of a function of x and a 
function of the other variables, the individual functions being 
mutually orthogonal, then these product functions are correct 
zeroth-order wave functions for this perturbation. This situa- 
tion arises whenever the unperturbed wave equation can be 
separated in a set of variables in which x is included. 

It may be pointed out that Equation 24-18 may also be written 
in the form 



ffn - W H l2 

#21 #22 ~ W 



Hex l 



H a 



H\a 
H 2 a 

H aa - W 



0, 



in which Ha = H% + H^ and W = W° k + W' kl , inasmuch as 
//§ is equal to Wl for i = j and to zero for i ^ j. This form is 
used in Section 30c. 

24a. An Example : Application of a Perturbation to a Hydrogen 
Atom. — As an illustration of the application of perturbation 
theory to degenerate systems, let us consider a hydrogen atom to 
which a perturbation which is a function of x only has been 
applied. Since the lowest state of the hydrogen atom is non- 
degenerate, the treatment of Section 23 applies to it and we have 
the result that 

W = Moof(x)dr 

with W = f(x). For the second energy state, however, we need 
to use the treatment for degenerate systems, since for W\ = — % 
Rhc there are four wave functions, 



^2. = ^20 



fa* = ^211 



\32raJ \a J 



cost?, 



VI-24a] FIRST-ORDER PERTURBATION THEORY 173 



**- - *- - aE 6 "" © • ^ e_i * sin *' 
**- = **» - >&'^) • ^ e+ * sin *» 

as given in Chapter V. In order to set up the secular equation 
for this system we need the integrals 

Even without specifying the form of the function /(x) further, 
we can say certain things about these integrals. Since the com- 
plex conjugate of e~~ lV is e+*> and e~ t >e +t "*' = 1, we see that 

" = "2lT,2lI = "211,211 

regardless of the nature of H', so long as it is real. By expressing 
x in polar coordinates through the equation 

x = r sin # cos <p, 

we see that f(x) is the same function of <p' = 2tt — <p as it is of <p, 
since cos (2tt — p) = cos <p. If we make this substitution in 
an integral over <p we get the result 

f 2 "g(<p)d<p = -£j(2* ~ <p'W = J^VZar - ^W = 

ffifrr - v)d<p, (24-21) 

since it is immaterial what symbol we use for the variable of 
integration in a definite integral. This substitution also changes 
e~ i<p into e-** 2 *-^ or e + * v ', so that by its use we can prove the 
identity 

D = "200.2U == -"200,211- 

f(x) is also unchanged in form by the substitution & = tt •— #', 
since sin (x — #') = sin #'. Also, we have the relation 

fVW sin &d$ = fW - #') sin #'<M' = 

r fif(x - t>) sin tfcW, (24-22) 



X' 



/o 

in which the factor sin # is introduced because it occurs in the 
volume element dr of polar coordinates. The substitution 
# = 7T — $' does not leave cos & unchanged, however, since 
cos (it — #') = — cos #'. By employing this substitution we 
can show that 



174 



PERTURBATION THEORY 



[VI-24a 



-"210,200 — ""-"210,200 



or 



ffsi 



= 0, 



'210,200 

since the integrand is unaltered by the substitution except for 
the cosine factor in ^§ 10 which changes sign. Similarly we 
find 



"210,211 "~ 



and Ho 



210,211 



= 0. 



Finally we have the general rule that 

TJ f — TT f * 

n 2lm,2Vm' ~ " 2l'm',2ltn' 

We are now in a position to write down the secular equation for 
this perturbation, using the relations we have obtained among 
the elements H' tlmtiVm >. It is (using the order 200, 211, 211, 210 
for the rows and columns) 

A-W D D 

D B -W E 

D E B - W 

C-W 



0. (24-23) 



The symbols A, B, etc., have the meanings: A = # 20 



B — /i2ii,211> C — H 2 



10,210) 



D =H' 



and E = HL 



We may obtain one root of this equation at once. Since the 
other elements of the row and the column which contains C — W 
are all zero, C — W is a factor of the determinant and may be 
equated to zero to obtain the root W = C. The other three 
roots may be obtained by solving the cubic equation which 
remains, but inspection of the secular equation suggests a simpler 
method. Determinants have the property of being unchanged 
in value when the members of any row are added to or sub- 
tracted from the corresponding members of any other row. The 
same is true of the columns. We therefore have 



A - 


W 


t 




D 


D 






D 




B - W 


E 




D 




E 


B -W 








1 


A-W 


2D 






"2 


D B 
D B 


-W'+E B -W - E 
-W' + E E - B + W 


1 


A 


— 


W 2D 







1 
~4 




2D 


2(5 + E - 


W) 


= C 












2(5 - 1 


E - W) 





(24-24) 



VI-24a] FIRST-ORDER PERTURBATION THEORY 175 

in which we have first added the last column to the second 
column to form a new second column and subtracted the last 
column irom the second column to form a new third column, and 
then repeated this process on the rows instead of the columns. 
The result shows that we have factored out another root, 
W' = B — E, leaving now the quadratic equation 

(A - W'KB + E - W) - 2Z> 2 = 

which determines the remaining two roots. 

The process by which we have factored the secular equation 
into two linear factors and a quadratic corresponds to using 
the real functions ^ 2 „ ^2 Px , fap,, and \f/ 2 p M for the ^'s instead 
of the set ^ 2 „ ^ 2P i, ^2 Pl , and ^ 2po (see Sec. 186). In terms of 
the real set the secular equation has the form 



A - W V2D 

V2D B + E - W 

B - E - W 

C - W 



= 0, 

(24-25) 



which, aside from the last row and column, differs from the last 
determinant of Equation 24-24 only by a constant factor. The 
proper zeroth-order wave functions for this perturbation are 
therefore ^ 2p „ ^ 2po and two linear combinations a^ 2 , + W 2p , 
and $yp2* — «^ 2px , in which the constants a and ft are determined 
by solving the quadratic factor of the secular equation, sub- 
stituting the roots into the equations for the coefficients of the 
linear combinations, and solving for the ratio a/fi. The 
normalization condition yields the necessary additional equation. 
It is to be noted that in place of ^ 2p „ and ^ 2p , any linear com- 
binations of these might have been used in setting up the secular 
equation 24-25, without changing the factoring of that equation, 
so that these linear combinations would also be satisfactory 
zeroth-order wave functions for this perturbation. 

Problem 24-1. Prove the statement of the last paragraph. 
Problem 24-2. Discuss the effect of a perturbation f(y) [in place of /(x)] 
on the system of Section 24a. 



176 PERTURBATION THEORY [VI-25 

25. SECOND-ORDER PERTURBATION THEORY 

In the discussion of Section 23 we obtained expressions for 
W and V in the series 

W = W° + \W + XW + • • • (25-1) 

and 

+ = xfr° + W + XV" + (25-2) 

In most problems it is either unnecessary or impracticable to 
carry the approximation further, but in some cases the second- 
order calculation can be carried out and is large enough to be 
important. This is especially true in cases in which the first- 
order energy W is zero, as it is for the Stark effect for a free 
rotator, a problem which is important in the theory of the meas- 
urement of dipole moments (Sec. 49/). 

The expressions for W" and ^" are obtained from the equation 
which results when the coefficient of X 2 in Equation 23-8 is put 
equal to zero and a solution obtained in a manner similar to 
that of the first-order treatment. We shall not give the details 
of the derivation but only state the results, which are, for the 
energy correction, 



in which 
and 



B' u = SWH'Mdr (25-4) 

HU = WH'Wdr (25-5) 



and the prime on 2 means that the term I = k is omitted. All 
other values of I must be included in the sum, however, including 
those corresponding to the continuous spectrum, if there is one. 
If the state W% is degenerate and the first-order perturbation 
has removed the degeneracy, then the functions to be used in 
calculating H' kl , etc., are the corrects zeroth-order functions found 
by solving the secular equation. 

If the energy level for the unperturbed problem is degenerate 
and the first-order perturbation does not remove the degeneracy > 
the application of the second-order correction will also not remove 
the degeneracy unless the term X 2 jff" is different from* zero, in 



VI-25a] SECOND-ORDER PERTURBATION THEORY 177 

which case the degeneracy may or may not be removed. The 
treatment in this case is closely similar to that of Section 24. 

25a. An Example : The Stark Effect of the Plane Rotator.— A 
rigid body with a moment of inertia I and electric moment 1 n, con- 
strained to rotate in a plane about an axis passing through its 
center of mass and under the influence of a uniform electric field 
E, is characterized by a wave equation of the form 2 

0+^OF + Kfl COS *)*=<>, 

in which <p is the angle of rotation. If we call — y.E cos <p the 
perturbation term, with E taking the place of the parameter X, 
then the unperturbed equation which remains when E = has 
the normalized solutions 

r m = -7^ eim *> ^ = 0, ±1, ±2, ±3, • • • , (25-6) 



and the energy values 



m 2 h 2 

Wi = |£- (26-7) 



In order to calculate the perturbation energy we shall need 
integrals of the type 

f*2ir r27r 

H' m m> = ~M J *t**t'COS^iV= -^1 ««*'-»>* cos **iV 

J"2t f2ic 

4ttJo 



= for m' ?± m ± 1, 

for m! = m ± 1. 



M ,_ _, _ „ ^ W (25-8) 

2 

Using this result we see at once that the first-order energy cor- 
rection is zero, for 

W' m = EH' mm = 0. (25-9) 

1 For a definition of /x see Equation 3-5. 

2 This equation can be obtained as the approximate wave equation for a 
system of two particles constrained by a potential function which restricts 
the particles to a plane and keeps them a fixed distance apart by an argu- 
ment similar to that used in the discussion of the diatomic molecule men- 
tioned in the footnote to Section 35c. 



178 PERTURBATION THEORY [VI-25a 

This problem is really a degenerate one, since W° depends only 
on \m\ and not on the sign of m, so that there are two wave 
functions for every energy level (other than the lowest). It is, 
however, not necessary to consider this circumstance in evaluat- 
ing W' m and WZ because neither the first- nor the second-order 
perturbation removes the degeneracy, and either the exponential 
functions 25-6 or the corresponding sine and cosine functions are 
satisfactory zeroth-order wave functions. 
The second-order energy, as given by Equation 25-3, is 

TJfftt _- JCT2 \"m,m— l) • F2 \"m,m+l) __ 4?T i/i A 

Wm "Wl- Wi_, + a Wl- W° m+1 h\lm* - 1)' 

(2&-10) 

so that the total energy, to the second order, is 

W , W o + XTr + v W „ , g^ + J£J*± _ . (25-11) 

It is interesting to point out the significance of this result 
in connection with the effect of the electric field on the polariza- 
bility of the rotator. The polarizability a is the proportionality 
factor between the induced dipole moment and the applied 
field E. The energy of an induced dipole in a field is then 
— %<xE 2 . From this and a comparison with Equation 25-11 we 
obtain the relation 

which shows that a is positive for m = 0; the induced dipole 
(whiqh in this case is due to the orienting effect of the field E on 
the permanent dipole m of the rotator) is therefore in the direction 
of the field E. For \m\ > 0, however, the opposite is true and 
the field tends to orient the dipole in the reverse direction. 

This is similar to the classical-mechanical result, which is 
that a plane rotator with insufficient energy to make a complete 
rotation in the field tends to be oriented parallel to the field 
while a rotator with energy great enough to permit complete 
rotation is speeded up when parallel and slowed down when 
antiparallel to the field so that the resulting polarization is 
opposed to the field. * 

1 An interesting application of perturbation theory has been made to the 
Stark effect of the hydrogen atom, the first-order treatment having been 



VI-25a] SECOND-ORDER PERTURBATION THEORY 179 

Problem 2&-1. Carry out a treatment similar to the above treatment for 
the rigid rotator in space, using the wave equation and wave functions found 
in the footnote of Section 35c. Discuss the results from the viewpoint of 
the last paragraph above. Compute the average contribution to the 
polarizability of all the states with given / and with m = — /, — I -f- 1> • • • > 
-j- /, assigning equal weights to the states in the averaging. 

given independently by Schrodinger, Ann. d. Phys. 80, 437 (1926), and 
P. S. Epstein, Phys. Rev. 28, 695 (1926), the second order by Epstein, loc. 
cit., G. Wentzel, Z. /. Phys. 38, 518 (1926), and I.Waller, ibid. 38, 635 (1926), 
and the third order by S. Doi, Y. Ishida, and S. Hiyama, Sci. Papers Tokyo 
9, 1 (1928), and M. A. El-Sherbini, Phil. Mag. 13, 24 (1932). See also 
Sections 27a and 27e. 



CHAPTER VII 

THE VARIATION METHOD AND OTHER 
APPROXIMATE METHODS 

There are many problems of wave mechanics which cannot be 
conveniently treated either by direct solution of the wave 
equation or by the use of perturbation theory. The helium 
atom, discussed in the next chapter, is such a system. No 
direct method of solving the wave equation has been found 
for this atom, and the application of perturbation theory is 
unsatisfactory because the first approximation is not accurate 
enough while the labor of calculating the higher approximations 
is extremely great. 

In many applications, however, there are methods available 
which enable approximate values for the energy of certain of the 
states of the system to be computed. In this chapter we shall 
discuss some of these, paying particular attention to the variation 
method, inasmuch as this method is especially applicable to the 
lowest energy state of the system, which is the state of most 
interest in chemical problems. 

26. THE VARIATION METHOD 

26a. The Variational Integral and Its Properties. — We shall 
show 1 in this section that the integral 

E = f<t>*H<t>dr (26-1) 

is an upper limit to the energy Wo of the lowest state of a system. 
In this equation, H is the complete Hamiltonian operator 

H[ 5—. -r-y q ) for the system under discussfon (Sec. 12a) and <f>(q) 
\Ziri o(j / f > ^ s . , ; 

is any normalized function of the coordinates of the system 

satisfying the auxiliary conditions of Section 9c for a satisfactory 

wave function. The function <t> is otherwise completely unre- 

1 C. Eckart, Phys. Rev. 36, 878 (1930). 

ISO 



VH-26al THE VARIATION METHOD 181 

stricted; its choice may be quite arbitrary, but the more wisely 
it is chosen the more closely will E approach the energy W . 

If we used for our function <t>, called the variation function, 
the true wave function \p of the lowest state, E would equal Wo) 
that is, 

E = MH+odr = Wo, (26-2) 

since 

/tyo = Wo4>o- 

If <f> is%ot equal to \po we may expand 4> in terms of the complete 
set of normalized, orthogonal functions ^ , ^1, • • * , ^n, • • * , 
obtaining 

<t> = %a n + n , with ^afan = 1. '■> (26^3) 

n n 

Substitution of this expansion in the integral for E -leads to the 
equation 

E = ZX<*n>J+:H+ n ,dT = %a:a n W n , (26^) 

n n' n 

inasmuch as the functions ^ n satisfy the equations 

Subtracting Wo, the lowest energy value, from both sides gives 
E - Wo = J -* ^* - TFo). (26^6) 

Since TF n is greater than or equal to Wo for all values of n and the 
coefficients a*a n are of course all positive oi* zero* the right side 
of Equation 26-6 is positive or zero. We have therefore proved 
that E is always an upper limit to Wo] that is, 

E^Wo. " (26-7) 

This theorem is the basis of the variation method for the 
calculation of the approximate value of the lowest energy level 
of a system. If we choose a number of variation functions 
^1, <t>2, $*,'•• and calculate the values Ei, EQE h • • • cor- 
responding to them, then each of these values of H is greater 
than the energy Wo, so that the lowest one is tlie nearest to Wo- 
Often the functions <t>i, fa, fa, • • • are only^distingui^^l by 
having different values of some parameter. The process of 



182 THE VARIATION METHOD lVH-2&a 

minimizing E with respect to this parameter may then be carried 
out in order to obtain the best approximation to Wo which the 
form of the trial function <t> will allow. 

If good judgment has been exercised in choosing the trial 
function <j> y especially if a number of parameters have been 
introduced into <f> in such a manner as to allow its form to be 
varied considerably, the value obtained for E may be very close 
to the true energy TP . In the case of the helium atom, for 
example, this method has been applied with great success, as is 
discussed in the next chapter. 

If E is equal to Wo then <t> is identical 1 with ^ (as can be 
seen from Eq. 26-6), so that it is natural to assume that if E is 
nearly equal to Wo the function 4> will approximate closely to 
the true wave function ^ . The variation method is therefore 
very frequently used to obtain approximate wave functions 
as well as approximate energy values. From Equation 26-6 
we see that the application of the variation method provides 
us with that function <j) among those considered which approxi- 
mates most closely to fo according to the following criterion: 
On expanding <j> — \f/ in terms of the correct wave functions ^ n , 

the quantity ^a n a n (W n — Wo) is minimized; that is, the sum 

n 

of the squares of the absolute values of the coefficients of the 
wave functions for excited states with the weight factors W n — W 
is minimized. For some purposes (as of course for the calcula- 
tion of the energy of the system) this is a good criterion to use ; 
but for others the approximate wave function obtained in this 
way might not be the most satisfactory one. 

Eckart 2 has devised the following way of estimating how 
closely a variation function approximates to the true solution ^ c 
by using E and the experimental values of Wo and W%. A very 
ieasonable criterion of the degree of approximation of <f> to ^o 
(for real functions) is the smallness of the quantity 

«-/(*- 4>o) 2 dr - /(** - 2^oX^n + 4>l)dr = 2 - 2a , 

(26-8) 

1 If the level Wo is degenerate, the equality of E and TT requires that 
<t> be |jpSitical with one of the wave functions corresponding to Wo. 
* Reference on p. 180. 



Vn-26a] THE VARIATION METHOD 183 

in which a is the coefficient of ^ in the expansion 26-3 of <t>. 
From Equation 26-6 we can write 

00 00 

E - Wo = %a* n (W n - Wo) } %a* n (Wx - Wo) 

n-0 n=l 

or 

E - Wo > (W x - Wo)(l - aj). 

Therefore if e 2 is small compared to €, we may combine this 
equation and Equation 26-8, obtaining 

^ E -Wo , ^ 1 E -W , „ n , 

€< f^¥ or l -" < 2WT=w; (26 " 9) 

Thus, from a knowledge of the correct energy values Wo and Wi 
for the two lowest levels of the systems and the energy integral E 
for a variation function <t>, we obtain an upper limit for the 
deviation of a from unity, that is, of the contribution to </> of 
wave functions other than i/'o. 

The variation method has the great drawback of giving only an 
upper limit to the energy, with no indication of how far from the 
true energy that limit is. (In Section 26e we shall discuss a 
closely related method, which is not, however, so easy to 
apply, by means of which both an upper and a lower limit can be 
obtained.) Nevertheless, it is very useful because there arise 
many instances in which we have physical reasons for believing 
that the wave function approximates to a certain form, and this 
method enables these intuitions to be utilized in calculating a 
better approximation to the energy than can be easily obtained 
with the use of perturbation theory. 

If we use for <j> the zeroth-order approximation to the wave 
function \pl discussed under perturbation theory, Chapter VI, 
and consider H as equal to H° + H', this method gives for 
E a value identical with the first-order perturbation energy 
Wl + W f o. If therefore we use for <f> a variation function con- 
taining parameters such that for certain values of the parameters 
<t> reduces to ^§, the value we obtain for E is always at least as 
good as that given by the first-order perturbation treatment. 
If 4> is set equal to the first-order wave function, the energy value 
E given by the variation method is the same, to the second 
power in the parameter X, as the second-order energy obtained 
by the perturbation treatment. 



184 THE VARIATION METHOD [VII-26b 

In case that it is not convenient to normalize <t>, the above 
considerations retain their validity provided that E is given 
by the expression 

* - *fi%r ^ 

26b. An Example : The Normal State of the Helium Atom. — 

In Section 236 we treated the normal state of the helium atom 
with the use of first-order perturbation theory. In this section 
we shall show that the calculation of the energy can be greatly 
increased in accuracy by considering the quantity Z which occurs 
in the exponent (p = 2Zr/a ) of the zeroth-order function given 
in Equations 23-34 and 23-37 as a parameter Z' instead of as a 
constant equal to the atomic number. The value of Z' is 
determined by using the variation method with <f> given by 

/ Z fz \ — Zri — Zrt 
4 = ^fo = ( — - )e «»e ao , (2&-11) 

yra / 

in which Z', the effective atomic number, is a variable parameter. 
In this problem, the Hamiltonian operator is 



H = - 



87r 2 m, 



- o <vl + vf)-z«{i + i) + £ 



in which Z is the true atomic number. The factors <j>i and <£ 2 
of <t> are hydrogenlike wave functions for nuclear charge Z'e, 
so that 0i satisfies the equation 

k2 "Vfri = ^~<t>i - Z'Wxfr (26-12) 



8ir*mo vrK1 . ri 

(W H being equal to e 2 /2a ) y with a similar equation for <£ 2 . Using 
these and the expression for H , we obtain 



E = -2Z'*W a + (Z' - Z)e 2 </>* - + - Udr + 



/*£+*> 



i 



<t>*—<t>dT. (26-13) 

T\2 



The first integral on the right has the value 



VII-26b] THE VARIATION METHOD 185 

j'3 f* °° C* C 2w \ 2Z ' ri 8Z' 3 e 2 C * 2Zri 

o Jo Jo Jo n a\ Jo 



2e 2 Z 

==-^ = 4Z'TF*. (26-14) 

The second integral of Equation 26-13 is the same as that of 
Equation 23-38 if Z is replaced by Z'. It therefore has the 
value 



/ 



<t>*—<i>dT = ^Z'Wh. (26-15) 

r i2 4 



Combining these results, we obtain for E the expression 

E = { -2Z' 2 + %Z' + 4Z'(Z' - Z)\W H . (2&-16) 
Minimizing E with respect to Z' gives 

||i = = (-4Z' + J + 8Z' - 4z)jF* 

or 

2' = Z - % 6 , (26-17) 

which leads to 

# = -2(Z - K*YWh. (26-18) 

As pointed out in Section 29c, this treatment cuts the error in 
the energy of helium to one-third of the error in the first-order 
perturbation treatment. In the same section, more elaborate 
variation functions are applied to this problem, with very 
accurate results. 

Problem 26-1. Calculate the energy of a normal hydrogen atom in & 
uniform electric field of strength F along the z axis by the variation method, 
and hence evaluate the polarizability a, such that the field energy is — J^aF 2 . 
Use for the variation function the expression 1 

1 The correct value of a for the normal hydrogen atom, given by the 
second-order perturbation theory (footnote at end of preceding chapter) is 

a = % a * = 0.667 • 10~ 2 * cm 8 . 

A value agreeing exactly with this has been obtained by the variation 
method by H. R. Hasse*, Proc. Cambridge Phil Soc. 26, 642 (1930), using the 
variation function ^i,(l + Az + Bzr). Hasse* also investigated the effects 
of additional terms (cubic and quartic) in the series, finding them to be 
negligible. The same result is given by the treatment of Section 27a. 



186 THE VARIATION METHOD [VII-26d 

*i.(l + Az) y 

minimizing the energy with respect to A, with neglect of powers of F higher 
than F 2 . 

a. = 4aJ = 0.59 • 10~ 24 cm 3 . Arts. 

26c. Application of the Variation Method to Other States. — 

The theorem E J> Wo, proved in Section 26a, may be extended 
in special cases to states of the system other than the lowest 
one. It is sometimes possible to choose <j> so that the first few 
coefficients a 0f a if • • • of the expansion 26-3 are zero. If, for 
example, a , ai, and a 2 are all zero, then by subtracting Wz 
from both sides of Equation 26-4 we obtain 

E-W z = ^a n a n *{W n - W*) > 0, (26-19) 

n 

since, although W - Wz, Wi - W*, and W 2 - Wz are negative, 
their coefficients are zero. In this case then we find the inequal- 
ity E J> Wz. 

There are several cases in which such a situation may arise. 
The simplest illustration is a one-dimensional problem in which 
the independent variable x goes from — oo to + °o and the 
potential function V is an even function of x, so that 

V(-x) = V(+x). 

The wave function belonging to the lowest level of such a system 
is always an even function; i.e., \po( — x) = foix); while ^i is odd, 
with \pi( — x) = —\l/\(x) (see Sec. 9c). If we therefore use for <j> 
an even function, we can only say that E is greater than or equal 
to Wo, but if 4> is an odd function, a will be zero (also all a n 's 
with n even) and the relation E ^ Wi will hold. For such a 
problem the variation method may be used to obtain the two 
lowest energy levels. 

The variation method may also be applied to the lowest state 
of given resultant angular momentum and of given electron-spin 
multiplicity, as will be discussed in the next chapter (Sec. 29d). 
Still another method of extending the variation method to levels 
other than the lowest is given in the following section. 

26d. Linear Variation Functions. 1 — A very convenient type of 
variation function is one which is the sum of a number of linearly 

x The generalized perturbation theory of Section 27a is closely related 
to the treatment discussed here. 



VII-26d] THE VARIATION METHOD 187 

independent functions xu X2, • • • , Xm with undetermined 
coefficients c h c 2 , • • • , c m . In other words the variation 
function <t> has the form 

<t> = CiXl + C2X2 + " * * + CmXm, (26-20) 

in which ci, c 2 , • • • , c m are the parameters which are to be 
determined to give the lowest value of E and therefore the best 
approximation to Wo. It is assumed that the functions xu 
X2, • • • , Xm satisfy the conditions of Section 9c. If we intro- 
duce the symbols 

H nn > = fxnHxn'dr and A nn , = JxnXn'dr, (26-21) 

in which for simplicity we have assumed that <t> is real, then the 
expression for E becomes 

m m 

/ . /* CnPn'tlnn' 
J<t>H<t>dT n = l n'-l (<>fr-<l<X\ 

E ~ i*w ~ ^T^ ( ' 

n-l n'-l 

or 

E ^ ^CnCn'Ann* = j) ^C n C n 'ffnn'. 
n n' n' 

To find the values of c h c 2 , • • • , c m which make E a minimum, 
we differentiate with respect to each c k : 

* n n N w n 

The condition for a minimum is that — - = for k = 1, 2, • • • , 

OCk 

m, which leads to the set of equations 

Xcn(H nk - AnkE) = 0, • * = 1, 2, • • • , m. (26-23) 

n 

This is a set of m simultaneous homogeneous linear equations in 
the m independent variables Ci, c 2 , • • • , c m . For this set of 



188 



THE VARIATION METHOD 



[VII-26d 



equations to have a non-trivial solution it is necessary that the 
determinant of the coefficients vanish (cf. Sec. 24); i.e., that 



i/21 



A21E 



Hn ~ Ai2# 

H 22 — A22E 



H mi — AmlE H m 2 — &m2.E 



H\ m 


- Ai JE? 


Him 


- A 2 »^ 


tl mm 


— A mm E 



= 0. 



(26-24) 



This equation is closely similar to the secular equation 24-17 of 
perturbation theory. It may be solved by numerical methods, 1 
or otherwise, and the lowest root E = Eq is an upper limit to 




Fia. 26-1. 



-Figure showing the interleaving of energy values for linear variation 
functions with added terms. 



the energy Wo- Substitution of this value of E in Equations 
26-23 and solution of these equations for c 2 , c 6 , • • • , c m in 
terms of ci (which can be used as a normalizing factor) gives the 
variation function <j>o corresponding to E . 

The other roots E h E 2 , • • • , E m -\ of Equation 26-24 are 
upper limits for Wi, Wi, • • • , TT m -i, respectively. 2 Further- 
more, it is possible to state how'these roots will be changed when 
a new trial function <£' is used, containing one more function 

Xm+l, 

<t>' = CiXl + C2X2 + • * * + CmXm + C m+ iXm+l. (26~25) 

In this case the roots E' 0y E[, E' 2) 
the old ones E Qy E h E 2 , • • 



• , 2^ will be separated by 
E m as shown in Figure 26-1. 



1 For a convenient numerical method see H. M. James and A. S. Coolidge, 
J. Chem. Phys. 1, 825 (1933). 

* J. K. L. MacDonald, Phys. Rev. 43, 830 (1933). 



Vn-26e] THE VARIATION METHOD 189 

In other words, the relations E' ^ E , E[ ^ E lt etc., and 
E ^ E[, Ei ^ E' 2 , etc., are satisfied. 

This method has proved to be very useful in practice, as will 
be illustrated by examples discussed in Chapters VIII and XII. 

The application of the variation method to wave mechanics grew from the 
work of Ritz, /. /. reine u. angew. Math. 135, 1 (1909), who considered the 
solution of certain differential equations by discussing the equivalent 
variation problem. It can be shown that a general normalized function fa 
which satisfies the boundary conditions of Section 9c and which makes the 
integral E — f<t>*HfadT a minimum relative to all variations in <£i is a 
solution of the differential equation H\p = W^, E then being equal to the 
corresponding characteristic energy value. A similar minimization of E 
with respect to all variations in another general normalized function fa 
with the added restriction that fa is orthogonal to <f>i leads to another solu- 
tion \p2 of the wave equation. By the continuation of this process of minimi- 
zation, all of the solutions can be found. Ritz proved that in certain cases 
a rigorous solution can be obtained by applying a limiting process to the 
integral /</>*#</>dr, in which <f> is represented as the sum of m functions of a 
convenient set of normalized orthogonal functions ^i, ^2, • • • which satisfy 
the boundary conditions, taken with arbitrary coefficients ci, C2, • • • , c». 
For each value of m the coefficients c m are determined so that the integral 
J<f>*H<f>dT is a minimum, keeping /</>*</>dr = 1. Ritz found that under 
certain restrictions the sequence of functions converges to a true solution 
of the wave equation and the sequence of values of the integral converges 
to the corresponding true characteristic value. The approximate method 
discussed in this section is very closely related to the Ritz method, differing 
from it in that the functions yp are not necessarily members of a complete 
orthogonal set and the limiting process is not carried out. 

Problem 26-2. Using a variation function of the form <f> — A -+• B cos 
<p -f- C sin <p, obtain an upper limit to the lowest energy level of the plane 
rotator in an electric field, for which the wave equation is 

6H Sr 2 I 

&i + ~~hT {W + * E cos ^ " °* 

26e. A More General Variation Method. — A method has been 
devised 1 which gives both an upper and a lower limit for an 
energy level. If we represent by E and D the integrals 

E = J<t>*H<t>dT and D = S(H4>)*(H4>)dr, (26-26) 

in which is a normalized trial variation function as before, then 
we shall show that some energy level Wk satisfies the relation 

E + VD - E 2 Z Wk} E - \/D - EK (26^-27) 

1 D. H. Wbinstbin, Proc. Nat, Acad, Sci. 20, 529 (1934); see also J. K. L. 
MacDonald, Phys, Rev, 46, 828 (1934). 



190 THE VARIATION METHOD [VII-26e 

To prove this we expand <t> as before (Eq. 26-3), so that 

E = ^afanWn, D = ^aZanWl and ^a*a n = 1. 

n n n 

(26-28) 
From this we obtain the result 

A = D - E* = %a£a n WZ - 2£ga>„F„ + E^a*a n = 

n n n 

X^a n (W n -E)\ (26-29) 

n 

There will be some energy level Wk which lies at least as near 
E as any other, i.e., for which 

(Wt - Ey <c (w n - E)\ 

Therefore A is related to Wk — E by the inequality 
A J> (Wk - E)^ata n 

n 

or 

A ^ (W k - E)\ (26-30) 

There are now two possible cases, 

Wk^E and W k < E. 

In the first case we have 

VA ^ W k - E, so that E + Va ^ W k ^ E; 

and in the second case 

VK >> E - W h and E > W k ^ E - VI. 

From this we see that the condition in Equation 26-27 applies 
to both cases. 

The application of this method to actual problems of the usual 
type is more difficult than that of the simple variation method 
because, in addition to the integral E, it is necessary to evaluate 
D, which ordinarily is considerably more difficult than E. 

It may be pointed out that by varying parameters in a function 
in such a way as to make A a minimum the function <j> is made to 
approach some correct wave function fa as closely as is permitted 
by the form of <j>. This method consequently may be considered 
as another type of variation method applicable to any state of a 
system. 



VII-27a] OTHER APPROXIMATE METHODS 191 

27. OTHER APPROXIMATE METHODS 

There are a number of other methods which may be used to 
obtain approximate wave functions and energy levels. Five of 
these, a generalized perturbation method, the Wentzel-Kramers- 
Brillouin method, the method of numerical integration, the 
method of difference equations, and an approximate second-order 
perturbation treatment, are discussed in the following sections. 
Another method which has been of some importance is based 
on the polynomial method used in Section 11a to solve the 
harmonic oscillator equation. Only under special circumstances 
does the substitution of a series for \[/ lead to a two-term recursion 
formula for the coefficients, but a technique has been developed 
which permits the computation of approximate energy levels for 
low-lying states even when a three-term recursion formula is 
obtained. We shall discuss this method briefly in Section 42c. 

27a. A Generalized Perturbation Theory. — A method of 
approximate (and in some cases exact) solution of the wave 
equation which has been found useful in many problems was 
developed by Epstein 1 in 1926, immediately after the publication 
of Schrodinger's first papers, and applied by him in the complete 
treatment of the first-order and second-order Stark effects of the 
hydrogen atom. The principal feature of the method is the 
expansion of the wave function in terms of a complete set of 
orthogonal functions which are not necessarily solutions of the 
wave equation for any unperturbed system related to the system 
under treatment, nor even necessarily orthogonal functions in 
the same configuration space. Closely related discussions of 
perturbation problems have since been given by a number of 
authors, including Slater and Kirkwood 2 and Lennard- Jones. 3 
In the following paragraphs we shall first discuss the method in 
general, then its application to perturbation problems and its 
relation to ordinary perturbation theory (Chap. VI), and finally 
as an illustration its application to the second-order Stark effect 
for the normal hydrogen atom. 

In applying this method in the discussion of the wave equation 

H+(x) = W+(x), (27-1) 

1 P. S. Epstein, Phys. Rev. 28, 695 (1926). 

* J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931). 

8 J. E. Lennard-Jones, Proc. Roy. Soc. A 129, 598 (1930). 



192 THE VARIATION METHOD [VII-27a 

in which x is used to represent all of the independent variables 
for the system, we express \l/(x) in terms of certain functions 
Fn(x), writing 

*(*) = %AnF n (x). (27-2) 

n 

The functions F n (x) are conveniently taken as the members of a 
complete set of orthogonal functions of the variables a;; it is not 
necessary, however, that they be orthogonal in the same con- 
figuration space as that for the system under discussion. Instead, 
we assume that they satisfy the normalization and orthogonality 
conditions 

JF*(x)F n (x)p(x)dx = 8 mn \ 
with 

flform = nA ( 27 ~ 3 ) 

for w^wj 



i 1 



in which p{x)dx may be different from the volume element dr 
corresponding to the wave equation 27-1. p{x) is called the 
weight factor 1 for the functions F n (x). On substituting the 
expression 27-2 in Equation 27-1, we obtain 

%A n (H - W)F n (x) = 0, (27-4) 

n 

which on multiplication by F*(x)p{x)dx and integration becomes 

%A n (H mn ~ Wbmn) = 0, TTl = 1, 2, • • • , (27-5) 

n 

in which 

#mn = fFZ(x)HF n (x)p(x)dx. (27-6) 

1 In case that the functions F n (x) satisfy the differential equation 



£{p(*)^} - «<*>* + *>(*)* m °> 



in which X is the characteristic-value parameter, they are known to form 
a complete set of functions which are orthogonal with respect to the weight 
factor p(x) . For a discussion of this point and other properties of differential 
equations of the Sturm-Liouviile type see, for example, R. Courant and 
D. Hilbert, "Methoden der mathematischen Physik," Julius Springer, 
Berlin, 1031. 



VH-27al 



OTHER APPROXIMATE METHODS 



193 



For an arbitrary choice of the functions F n (x) Equation 27-5 
represents an infinite number of equations in an infinite number 
of unknown coefficients A n . Under these circumstances ques- 
tions of convergence arise which are not always easily answered. 
In special cases, however, only a finite number of functions 
F n (x) will be needed to represent a given function ^(x) ; in these 
cases we know that the set of simultaneous homogeneous linear 
equations 27-5 has a non-trivial solution only when the deter- 
minant of the coefficients of the i n 's vanishes; that is, when the 
condition 



Hn H22 ~~ W Hiz 
H31 H 32 i/33 — W 



= (27-7) 



is satisfied. We shall assume that in the infinite case the mathe- 
matical questions of convergence have been settled, and that 
Equation 27-7, involving a convergent infinite determinant, is 
applicable. 

Our problem is now in principle solved: We need only to eval- 
uate the roots of Equation 27-7 to obtain the allowed energy 
values for the original wave equation, and substitute them in 
the set of equations 27-5 to evaluate the coefficients A n and 
obtain the wave functions. 

The relation of this treatment to the perturbation theory of 
Chapter VI can be seen from the following arguments. If the 
functions F n (x) were the true solutions ^»(x) of the wave equation 
27-1, the determinantal equation 27-7 would have the form 



Wi-W 

W 2 - W 

Wz - W 



= 0, (27-8) 



with roots W = W\ f W = W 2 , etc. Now, if the functions 
F n (x) closely approximate the true solutions rp n (x), the non- 
diagonal terms in Equation 27-7 will be small, and as an approxi- 
mation we can neglect them. This gives 

TTi = ffn, 
W 2 = #22, 

Wz = #33> 

etc., 



(27-9) 



194 



THE VARIATION METHOD 



[Vn-27a 



which corresponds to ordinary first-order perturbation theory, 
inasmuch as, if # can be written as H° + #', with 

H°F n (x) = W°F n (x), 

then W n = H nn has the value W n = W° n + JFZ(x)H'F n (x)p(x)dx, 
which is identical with the result of ordinary first-order per- 
turbation theory of Section 23 when p{x)dx = dr. Equation 
27-9 is more general than the corresponding equation of first- 
order perturbation theory, since the functions F n (x) need not 
correspond to any unperturbed system. On the other hand, 
it may not be so reliable, in case that a poor choice of functions 
F n (x) is made; the first step of ordinary perturbation theory is 
essentially a procedure for finding suitable zeroth-order functions. 
It may happen that some of the non-diagonal terms are large 
and others small; in this case neglect of the small terms leads to 
an equation such as 



#n - W #12 

#21 #22 - W 

#33 - W 

#44 - W 



which can be factored into the equations 



= o, 



ffll 



w 



H 2 



Ha 

#22 - W 


= 0, 


H 33 - W 


= o, 


H it - W 


= 0, 


etc. 





(27-10) 



It is seen that this treatment is analogous to the first-order 
perturbation treatment for degenerate states as given in Section 
24. The more general treatment now under discussion is espe- 
cially valuable in case that the unperturbed levels are not exactly 
equal, that is, in case of approximate degeneracy. 

A second approximation to the solution of Equation 27-7 
can be made in the following manner. Suppose that we are 
interested in the second energy level, for which the value #22 
is found for the energy as a first approximation. We introduce 
this expression for W everywhere except in the term #22 — W, 



Vn-27a] 



OTHER APPROXIMATE METHODS 



195 



and neglect non-diagonal terms except H 2n and //» 2 , thus obtain- 
ing the equation 



Hu — H22 H12 

H21 H22 ~ W H23 H24 

il 32 2/33 ~" H22 

H42 Ha — H22 



= 0. 

(27-11) 



On multiplying out the determinant, we convert this equation 
into the form 



(#22 - >WKH n - H 2i )(H 33 - HnHHu 

— i/i2#2l(//33 ~ H22) {Hu — H22) ' ' ' 

— Hz2H2z(H\\ — H22)(Hu — H22) • • * — 



H22) 



with the solution 



W 



H » ~ 2 7/~ 



H21H12 



H 2 



0, 



(27-12) 



in which the prime indicates that the term with I = 2 is omitted. 
This is analogous to (and more general than) the second-order 
perturbation treatment of Section 25; Equation 27-12 becomes 
identical with Equation 25-3 when Hu is replaced by Wi and 
H %1 by H' kl . 

Higher approximations can be carried out by obvious exten- 
sions of this method. If Equation 27-7 can be factored into 
equations of finite degree, they can often be solved accurately by 
algebraic or numerical methods. 

Let us now consider a simple example, 1 the second-order 
Stark effect of the normal hydrogen atom, using essentially the 
method of Epstein (mentioned above). This will also enable 
us to introduce and discuss a useful set of orthogonal functions. 

The wave equation for a hydrogen atom in an electric field 
can be written as 



8tt 2 m 



V 2^ _ L.^ + eFz ^ = w ^ f 



(27-13) 



in which eFz represents the interaction with an electric field of 
strength F along the z axis. In order to discuss this equation we 
shall make use of certain functions /^x M (£, #, <p), defined in terms 

1 The study of this example can be omitted by the reader if desired. 



196 THE VARIATION METHOD [VH-27a 

of the associated Laguerre and Legendre functions (Sees. 19 
and 20) as 

*Wtt, *, <p) = A*(QO*(mM, (27-14) 

in which 

A *<«> = [ferw'T^'^"'' (27_15) 

LJ+j^U) being an associated Laguerre polynomial as defined in 
Section 206. The functions 6\ M (#) and $ M (^) are identical with 
the functions 6/m(#) and $ m (<p) of Equations 21-2 and 21-3 
except for the replacement of I and m by X and /z. It is found 
by the use of relations given in Sections 19 and 20 that F v \^ } #, <p) 
satisfies the differential equation 



W 4- ? §F (v _ 1 \ F 1 dW 

dp f a? \f V £ 2 sin 2 a^ 2 






+ £ 2 sin 1 

The functions are normalized and mutually orthogonal with 
weight factor £ , satisfying the relations 

(v = v' 
JTJCJC >WWM; sin MM, = 1 for X « X'( 

= otherwise. 

If we identify £ with 2Zr/n'a , where a = h 2 /^K 2 ixe 2 7 then 
the functions F„\ M become identical with the hydrogen-atom wave 
functions \pnim for the value n = n' of the principal quantum 
number n, but not for other values of n; the functions F„x M all 
contain the same exponential function of r, whereas the hydrogen- 
atom wave functions for different values of n contain different 
exponential functions of r. 

For the problem at hand we place n' equal to 1 and Z equal to 1, 
writing 

2r _ h 2 

The functions F„x M then satisfy the equation 

V'F, Xm + Q - 0F,x M - (y = 1} * W (27-19) 



* - -> «o - 33-5- (27-18) 



Vn-27a] OTHER APPROXIMATE METHODS 

Now let us write our wave equation 27-13 as 



vv + 



in which 



(H>- 



A ~ 4e' 



A£co8&\f/ = fi\py 



= 



Wa 
' 2e 2 



1 

i'J 



197 



(27-20) 



(27-21) 



and the operation V 2 refers to the coordinate £ rather than r, 
£ being given by Equation 27-18. To obtain an approximate 
solution of this equation in terms of the functions F^ we shall 
set up the secular equation in the form corresponding to. second- 
order perturbation theory for the normal state, as given in 
Equation 27-11; we thus obtain the equation 



"21 H22 
H*i Hm 



= 0, (27-22) 



in which 



Hi, = jj Jf? (v 2 +t - i - M«* 



* 



•) 



F£*d( sin UUip, (27-23) 

i and j being used to represent the three indices v, X, /x. The 
factor 2 before p arises from the fact that the functions F„\p 
are not normalized to unity with respect to the volume element 
£ 2 d£ sin $d#d<p. 

It is found on setting up the secular equation 27-22 that only 
the three functions F100, ^210, and F310 need be considered, inasmuch 
as the equation factors into a term involving these three func- 
tions only (to the degree of approximation considered) and terms 
involving other functions. The equations 



and 



£ 2 cos &F 100 = 4\/2F 21 o - 2\Z2Fz 



(27-24) 



€**, - ~ {(" - *)(» + X + 1)}^+i.xm + 2vF,xm - 

{ (v + \)(v - X - 1) } W,-^, (27-25) 



198 THE VARIATION METHOD [VII-27b 

together with Equation 27-17, enable us to write as the secular 
equation for these three functions 

-2/3 -4\/2A 2V2A 
-4V2A -1 =0. (27-26) 

2y/2A -2 

The root of this is easily found to be fi = ISA 2 , which corresponds 

to 

e 2 9 

or 

W" = W -W° = -V^W 2 ' (27-27) 

This corresponds to the value 

a = y 2 a\ = 0.677 • 10-" cm 3 
for the polarizability of the normal hydrogen atom. 

Problem 27-1. Derive the formulas 27-24 and 27-25. 

Problem 27-2. Discuss the first-order and* second-order Stark effects 
for the states n = 2 of the hydrogen atom by the use of the functions F„xm- 
Note that in this case the term in A can be neglected in calculating 

tl v'\' n' ,v' '\' V" 

unless v or v" is equal to 2, and that the secular equation can be factored 
into terms for p. = -\-l, n =0, and \x — — 1, respectively. 

27b. The Wentzel-Kramers-Brillouin Method. — For large 
values of the quantum numbers or of the masses of the particles 
in the system the quantum mechanics gives results closely similar 
to classical mechanics, as we have seen in several illustrations. 
For intermediate cases it is found that the old quantum theory 
often gives good results. It is therefore pleasing that there 
has been obtained 1 an approximate method of solution of the 
wave equation based on an expansion the first term of which 
leads to the classical result, the second term to the old-quantum- 
theory result, and the higher terms to corrections which bring 
in the effects characteristic of the new mechanics. This method 
is usually called the Wentzel-Kramers-Brillouin method. In 
our discussion we shall merely outline the principles involved 
in it. 

1 G. Wentzel, Z. /. Phys. 38, 518 (1926); H. A. Kramers, Z.f. Phys. 39, 
828 (1926); L. Brillouin, J. de phys. 7, 353 (1926); J. L. Dunham, Phyg. 
Rev. 41, 713 (1932). 



VH-27b] OTHER APPROXIMATE METHODS 199 

For a one-dimensional problem, the wave equation is 

g + ^ar-™ = o, -oo<*< + oo. 

If we make the substitution 

+ = ffi vd \ (27-28) 

we obtain, as the equation for y, 

A g = 2m (W -V)-y* = p>- y\ (27-29) 



in which p = ±\/2m(W — V) is the classical expression for 
the momentum of the particle. We may now expand y in powers 
of h/2iri } considering it as a function of h, obtaining 

y - f • + £* + (as)* + ' ' • • (27 " 30) 

Substituting this expansion in Equation 27-29 and equating the 
coefficients of the successive powers of h/2iri to zero, we obtain 
the equations 



Vo 



= p = ±V2m(W - V), (27-31) 

yi = -iio == ~~k = 4(tt - vy - (27 ~ 32) 

y2 = ~^ 2 f5F' 2 +4F"(lf - 7)}(2ro)-H(TP - F)-* (27-33) 

dF rf 2 F 

in which F ; = ^- and V" = ^4- 
aa; ax 2 

The first two terms when substituted in Equation 27-28 lead 
to the expression 



1 2«i| 



* S W - V) *e a" J^™* (27-34) 

as an approximate wave function, since 

J »i& = |J ^zrp^ = + ij wzrv = -\ lo s W ^ F ) 

so that 

The probability distribution function to this degree of approxima- 
tion is therefore 

*** = NKW - F)-H = const, -i (27-35) 



200 THE VARIATION METHOD lVH-27b 

agreeing with the classical result, since p is proportional to the 
velocity and the probability of finding a particle in a range dx is 
inversely proportional to its velocity in the interval dx. 

The approximation given in Equation 27-34 is obviously not 
valid near the classical turning points of the motion, at which 
W = V. This is related to the fact that the expansion in 
Equation 27-30 is not a convergent series but is only an asymp- 
totic representation of y, accurate at a distance from the points 
at which W = V. 

So far nothing corresponding to quantization has appeared. 
This occurs only when an attempt is made to extend the wave 
function beyond the points W = V into the region with W less 
than V. It is found 1 that it is not possible to construct an 
approximate solution in this region satisfying the conditions 
of Section 9c and fitting smoothly on to the function of Equation 
27-34, which holds for the classically allowed region, unless W 
is restricted to certain discrete values. The condition imposed 
on W corresponds to the restriction 

fydx = nh, n = 0, 1, 2, 3, • • • , (27-36) 

in which the integral is a phase integral of the type discussed in 
Section 56. If we insert the first term of the series for y, y = p, 
we obtain the old-quantum-theory condition (Sec. 56) 

fpdx = nh, n = 0, 1, 2, 3 • • • . (27-37) 



For systems of the type under discussion, the second term 

h 
5i* 



introduces half -quantum numbers; i.e., with y = y + 75-.2/1, 



(bydx = Spdx + ~—.<£yidx = (Lpdx — ^ = nh, 

so that 

fpdx = (n + V 2 )h (27-38) 

to the second approximation. (The evaluation of integrals such 
as jfyidx is best carried out by using the methods of complex 
variable theory, which we shall not discuss here. 2 ) 

This method has been applied to a number of problems and 
is a convenient one for many types of application. Its main 

1 Even in its simplest form the discussion of this point is too involved to 
be given in detail here. 

* J. L. Dunham, Phy* Rev. 41, 713 (1932). 



VH-27C] OTHER APPROXIMATE METHODS 201 

drawback is the necessity of a knowledge of contour integration, 
but the labor involved in obtaining -the energy levels is often 
considerably less than other methods require. 

27c. Numerical Integration. — There exist well-developed meth- 
ods 1 for the numerical integration of total differential equations 
which can be applied quite rapidly by a practiced investigator. 
The problem is not quite so simple when it is desired to find 
characteristic values such as the energy levels of the wave equa- 
tion, but the method is still practicable. 

Hartree, 2 whose method of treating complex atoms is dis- 
cussed in Chapter IX, utilizes the following procedure. For 
some assumed value of W, the wave equation is integrated 
numerically, starting with a trial function which satisfies the 
boundary conditions at one end of the range of the independent 
variable x and carrying the solution into the middle of the range 
Another solution is then computed for this same value of W, 
starting with a function which satisfies the boundary conditions 
at the other end of the range of x. For arbitrary values of W 
these two solutions will not in general join smoothly when they 
meet for some intermediate value of x. W is then changed 
by a small amount and the process repeated. After several 
trials a value of IF is found such that the right-hand and left- 
hand solutions join together smoothly (i.e., with the same 
slope), giving a single wave function satisfying all the boundary 
conditions. 

This method is a quantitative application of the qualitative 
ideas discussed in Section 9c. The process of numerical integra- 
tion consists of starting with a given value and slope for ^ at a 
point A and then calculating the value of ^ at a near-by point B 

by the use of values of the slope and curvature -7-5 at A, the latter 

being obtained from the wave equation. 

This procedure is useful only for total differential equations in 
one independent variable, but there are many problems involving 
several independent variables which can be separated into total 

1 E. P. Adams, "Smithsonian Mathematical Formulae," Chap. X, The 
Smithsonian Institution, Washington, 1922; E. T. Whittaker and G. 
Robinson, "Calculus of Observations," Chap. XIV, Blackie and Son., Ltd., 
London, 1929. 

1 D. R. Hartree, Proc. Cambridge Phil. Soc. 24, 105 (1928); Mem. 
Manchester Phil. Soc. 77, 91 (1932-1933). 



202 



THE VARIATION METHOD 



[VII-27d 



differential equations to which this method may then be applied. 
Hartree's method of treating complicated atoms (Sec. 32) and 
Bureau's calculation of the energy of HJ (Sec. 42c) are illustra- 
tions Oa the use of numerical integration. 

27d. Approximation by the Use of Difference Equations.— The 
wave equation 



g + k 2 (W - V)+ = o, 



_8x 2 m 
10 " h 2 ' 



(27-39) 



may be approximated by a set of difference equations, 1 



±(fc-! - 2h + * m ) + k 2 {W - 7(x,)}*< = 0, (27-40) 



or 



5)m, = w^„ 



(27-41) 



in which ^i, ^ 2 , 



, *<, 



function ^ at the points Zi, x 2 , 



are numbers, the values of the 
■••,£»,••• uniformly spaced 




Fiq. 27-1. — The approximation to a wave function ¥ by segments of straight lines. 

along the x axis with a separation x» — s*-i = a. To prove this 
we consider the approximation to \p formed by the polygon of 
straight lines joining the points (x\> ^i), (z 2 , ^2), • • • > 
(s», ^*), • • • of Figure 27-1. The slope of ^ at the point 
halfway between Xi-i and Xi is approximately equal to the slope 
of the straight line connecting x^i and x i} which is (^ — ^;_i)/a. 
The second derivative of ^ at x = Xi is likewise approximated by 
1/a times the change in slope from (xi + Xi-i)/2 to (a;* + Xi+i)/2; 
that is, 

dx 2 a 2 



(27-42) 



1 R. G. D. Richardson, Trans. Am. Math. Soc. 18, 489 (1917); R. Courant, 

\C ESannivnTriTra anH "FT T.mwv Mnthrmntijirh* Ainnnlein Iftft 39 (1 Q9ft"\ 



VH-27d] OTHER APPROXIMATE METHODS 203 

The differential equation 27-39 is the relation between the 

curvature -^ata point and the function k 2 (W — V)^ at that 

point, so that we may approximate to the differential equation 
by the set of equations 27-40, there being one such equation for 
each point x». The more closely we space the points x if the more 
accurately do Equations 27-40 correspond to- Equation 27-39. 

Just as the lowest energy W of the differential equation can be 
obtained by minimizing the energy integral E = f<t>*H<t>dT with 
respect to the function <£, keeping /</>*</>dr = 1, so the lowest 
value of W giving a solution of Equations 27-40 may be obtained 
by minimizing the quadratic form 

2^bi } (t>i<t>j 
E = -2— -, (27-43) 

X 

in which to, to, • • • , 4>i, • • • are numbers which are varied 
until E is a minimum. (Just as <t> must obey the boundary 
conditions of Section 9c, so the numbers # t must likewise approxi- 
mate a curve which is a satisfactory wave function.) 

A convenient method 1 has been devised for carrying out this 
minimization. A set of trial values of <£» is chosen and the 
value of E is calculated from them. The true solutions \p i} 
to which the values of to will converge as we carry out the 
variation, satisfy Equations 27-40. Transposing one of these 
gives 

*' = 2 - aW[W - Viz*)}' (27_44) 

If the (fc's we choose are near enough to the true values ^», then 
it can be shown 1 that, by putting to-i and </> 1+ i in place of \pi-i 
and yj/i+i and E in place of W in Equation 27-44, the resulting 
expression gives an improved value ^ for <j> i} namely, 

In this way a new set <t>[, to, • • • , <£•, • • • can be built up 
from the initial set fa, to, - • - , <t>i, • - • t ^e new set giving a 

1 G. E. Kimball and G. H. Shortley, Phys. Rev. 45, 815 (1934). 



204 THE VARIATION METHOD [VII-27e 

lower and therefore a better value of E than the first set. This 
process may then be repeated until the best set <£» is obtained 
and the best value of E. 

This procedure may be modified by the use of unequal intervals, 
and it can be applied to problems in two or more dimensions, 
but the difficulty becomes much greater in the case of two 
dimensions. 

Problem 27-3. Using the method of difference equations with an 
interval a = }4, obtain an upper limit to the lowest energy Wo and an 
approximation to ^ for the harmonic oscillator, with wave equation 

^ + (X - x*)+ = (see Eq. 11-1). 

27e. An Approximate Second-order Perturbation Treatment. 
The equation for the second-order perturbation energy (Eq. 25-3) 
is 

w '»' - 2 'w~ff + H "» ( 27 - 46) 

i 

with 

H' u = M*H'+fdT 
and 

The sum may be rearranged in such a manner as to permit an 
approximate value to be easily found. On multiplying by 

W + W ' becomes 

Wl^ kl lk "*" ^J Wl (W° k - wtf 
I I 

Now we can replace 1 ^'H' kl H' lk by (H' 2 ) kk - {H f kk )\ obtaining 

i 

1 To prove this, we note that H'ypl = £jH' lk \l/\ (as is easily verified by 

I 
multiplication by \p t * and integration). Hence 

(#")** - f+l'H'VldT - f+l*H'H'+ldr = fr k *H'(%H' lk +tydT 
The sum JJ « #m#j* differs from this only by the term with I - A;, (H'h)*. 



VII-27e] OTHER APPROXIMATE METHODS 205 
for W' k ' the expression 

W" - (#' ')** (H'kk) 2 , Tin , N? Wj #;»#!* . (<Y7_A7\ 

Wk " "W! WT + "** + ^j W2 (W2 - Wf) { } 



in which 



(ff'% = M*H'V k dr. (27-48) 

This expression is of course as difficult to evaluate as the 
original expression 27-46. However, it may be that the sum is 
small compared with the other terms. For example, if k repre- 
sents the normal state of the system, and the origin for energy 
measurements is such that Wl is negative, the terms in the sum 
will be negative, for W® negative, and positive for W? positive, 
and there may be considerable cancellation. It must be empha- 
sized that the individual terms in this expression are dependent 
on the origin chosen for the measurement of energy (the necessity 
for an arbitrary choice of this origin being the main defect of the 
approximate treatment we are describing). If this origin were 
to be suitably chosen, this sum could be made to vanish, the 
second-order perturbation energy then being given by integrals 
involving only one unperturbed wave function, that for the 
state under consideration. The approximate treatment consists 
in omitting the sum. 

As an example let us take the now familiar problem of the 
polarizability of the normal hydrogen atom, with H' = eFz. 
We know that H' uu vanishes. The integral (H' 2 ) u ,u is equal 
to e 2 F 2 (z 2 ) u , U) and, inasmuch as r 2 = x 2 + y 2 + z 2 and the 
wave function for the normal state is spherically symmetrical, 
the value of (z 2 )i,,u is just one-third that of (r 2 ) UfU , given in 
Section 21c as 3ag. Thus we obtain 

e 2 F 2 al 
\n" = ° 

Wl 

If we use the value — e 2 /2a for Wl, (taking the ionized atom at 
zero energy), we obtain 

W" = -2F 2 a 3 , 

which corresponds to the value a = 4aJ for the polarizability. 
This is only II per cent less than the true value (Sec. 27a), 
being just equal to the value given by the simple treatment of 
Problem 26-1. 



206 THE VARIATION METHOD [Vn-27e 

It is interesting to note that if, in discussing the normal state 
of a system, we take as the zero of energy the first unperturbed 
excited level, then the sum is necessarily positive and the approxi- 
mate treatment gives a lower limit to W". In the problem of the 
normal hydrogen atom this leads to 

giving for a the upper limit x %al f which is 18 per cent larger 
than the correct value %a%. Inasmuch as the value a = 4a§ given 
by the variation method is a lower limit, these two very simple 
calculations fix a to within a few per cent. 

It was pointed out by Lennard- Jones 1 that this approximate 
treatment of W' k ' corresponds to taking as the first-order per- 
turbed wave function the approximate expression (not normalized) 

** = «(1 + AH' + • • • ) (27-49) 

in which A = 1/Wl 

This suggests that, when practicable, it may be desirable to 
introduce the perturbation function in the variation function 
in this way in carrying out a variation treatment. Examples of 
calculations in which this is done are given in Sections 29# 
and 47. 

1 J. E. Lennard- Jones, Proc. Roy. Soc. A129, 598 (1930). 



CHAPTER VIII 

THE SPINNING ELECTRON AND THE PAULI 

EXCLUSION PRINCIPLE, WITH A DISCUSSION OF THE 

HELIUM ATOM 

28. THE SPINNING ELECTRON 1 

The expression obtained in Chapter V for the energy levels of 
the hydrogen atom does not account completely for the lines 
observed in the hydrogen spectrum, inasmuch as many of the 
lines show a splitting into several components, corresponding 
to a fine structure of the energy levels not indicated by the simple 
theory. An apparently satisfactory quantitative explanation 
of this fine structure was given in 1916 by the brilliant* work of 
Sommerfeld, 2 who showed that the consideration of the rela- 
tivists change in mass of the electron caused the energy levels 
given by the old quantum theory to depend to some extent on 
the azimuthal quantum number k as well as on the total quantum 
number n, the splitting being just that observed experimentally 
not only for hydrogen and ionized helium but also for x-ray 
lines of heavy atoms. This explanation was accepted for 
several years. Shortly before the development of the quantum 
mechanics, however, it became evident that there were trouble- 
some features connected with it, relating in particular to the 
spectra of alkalilike atoms. A neutral alkali atom consists in 
its normal state of an alkali ion of particularly simple electronic 
structure (a completed outer group of two or eight electrons) 
and one valence electron. The interaction of the valence electron 
and the ion is such as to cause the energy of the atom in various 
quantum states to depend largely on the azimuthal quantum 
number for the valence electron as well as on its total quantum 
number, even neglecting the small relativistic effect, which ip 
negligible compared with the electron-ion interaction. How- 

1 For a more detailed treatment of this subject see L. Pauling and S. 
Goudsmit, "The Structure of Line Spectra," Chap. IV. 

2 A. Sommerfeld, Ann. d. Phys. 51, 1 (1916). 

207 



208 THE SPINNING ELECTRON [Vm-28 

ever, the levels corresponding to given values of these two 
quantum numbers were found to be often split into two levels, 
and it was found that the separations of these doublet levels are 
formally representable by the Sommerfeld relativistic equation. 
Millikan and Bowen 1 and Land6, 2 who made this discovery, 
pointed out that it was impossible to accept the relativistic 
mechanism in this case, inasmuch as the azimuthal quantum 
number is the same for the two components of a doublet level, 
and they posed the question as to the nature of the phenomenon 
involved. 

The answer was soon given by Uhlenbeck and Goudsmit, 3 
who showed that the difficulties were removed by attributing to 
the electron the new properties of angular momentum and 
magnetic moment, such as would be associated with the spinning 
motion of an electrically charged body about an axis through 
it. The magnitude of the total angular momentum of the 



electron is y/s(s + l)^-> in which s, the electron-spin quantum 

number, is required by the experimental data to have the value 
3^. The component of angular momentum which the electron 

spin can possess along any prescribed axis is either +rr- or 

-sri that is, it is given by the expression m,~-, in which the 

quantum number m, can assume only the values +}4 an d —}i* 
To account for the observed fine-structure splitting and Zeeman 
effects it is found that the magnetic moment associated with 
the electron spin is to be obtained from its angular momentum 
by multiplication not by the factor e/2woc, as in the case of 
orbital magnetic moment (Sec. 21d), but by twice this factor, 
the extra factor 2 being called the* Landi g factor for electron spin. 
In consequence the total magnetic moment of the electron spin 



1 R. A. Millikan and I. S. Bowen, Phys. Rev. 24, 223 (1924). 

* A. Landb, Z. /. Phys. 25, 46 (1924). 

* G. E. Uhlenbeck and S. Goudsmit, Naiurwis&enschaflen 13, 953 (1925); 
Nature 117, 264 (1926). The electron spin was independently postulated by 
R. Bichowsky and H. C. Urey, Proc. Nat. Acad. Set. 12, 80 (1926) (in whose 
calculations there was a numerical error) and had been previously suggested 
on the basis of unconvincing evidence by several people. 



Vm-281 THE SPINNING ELECTRON 209 

is 2~ o~\l- • - or a/3 Bohr magnetons, and the component along 

a prescribed axis is either +1 or — 1 Bohr magneton. 

It was shown by Uhlenbeck and Goudsmit and others 1 that the 
theory of the spinning electron resolves the previous difficulties, 
and the electron spin is now accepted as a property of the electron 
almost as well founded as its charge or mass. The doublet 
splitting for alkalilike atoms is due purely to the magnetic 
interaction of the spin of the electron and its orbital motion. 
The fine structure of the levels of hydrogenlike atoms is due to 
a particular combination of spin and relativity effects, resulting 
in an equation identical with Sommerfeld's original relativistic 
equation. The anomalous Zeeman effect shown by most atoms 
(the very complicated splitting of spectral lines by a magnetic 
field) results from the interaction of the field with both the 
orbital and the spin magnetic moments of the electrons, the 
complexity of the effect resulting from the anomalous value 2 
for the g factor for electron spin. 2 

The theory of the spinning electron has been put on a particu- 
larly satisfactory basis by the work of Dirac. In striving to 
construct a quantum mechanics compatible with the require- 
ments of the theory of relativity, Dirac 3 was led to a set of 
equations representing a one-electron system which is very 
different in form from the non-relativistic quantum-mechanical 
equations which we are discussing. On solving these, he found 
that the spin of the electron and the anomalous g factor 2 were 
obtained automatically, without the necessity of a separate 
postulate. The equations led to the complete expression for 
the energy levels for a hydrogenlike atom, with fine structure, 
and even to the foreshadowing of the positive electron or positron, 
discovered four years later by Anderson. 

So far the Dirac theory has not been extended to systems 
containing several electrons. Various methods of introducing 

1 W. Pauli, Z. f. Phys. 36, 336 (1926); W. Heisenberg and P. Jordan, 
Z. /. Phys. 37, 266 (1926); W. Gordon, Z. f. Phys. 48, 11 (1928); C. G. 
Darwin, Proc. Roy. Soc. A 118, 654 (1928); A. Sommerfeld and A. Unsold, 
Z. f. Phys. 36, 259; 38, 237 (1926). 

'For a fuller discussion see Pauling and Goudsmit, "The Structure of 
line Spectra," Sees. 17 and 27. 

* P. A. M. Dirac, Proc. Roy. Soc. A117, 610; A118, 351 (1928). 



210 THE SPINNING ELECTRON [VHI-29a 

the spin in non-relativistic quantum mechanics have been 
devised. Of these we shall describe and use only the simplest 
one, which is satisfactory so long as magnetic interactions 
are neglected, as can be done in treating most chemical and 
physical problems. This method consists in introducing a 
spin variable w, representing the orientation of the electron, 
and two spin wave functions, a(o>) and /3(w), the former cor- 
responding to the value +J^ for the spin-component quantum 
number m a (that is, to a component of spin angular momentum 
along a prescribed axis in space of +} / 2h/2r) and the latter to 
the value — \^ f or m «- The two wave functions are normalized 
and mutually orthogonal, so that they satisfy the equations 

/a 2 (co)da> = 1, ) 

J*0 2 (co)dco = 1, \ (28-1) 

Ja(co)j8(co)doj = 0.) 

A wave function representing a one-electron system is then 
a function of four coordinates, three positional coordinates 
such as x, y, and z, and the spin coordinate w. Thus we write 
$(%, V, 2)<*(co) and \f/(x, y, z)/3(co) as the two wave functions cor- 
responding to the positional wave function \[/(x, y, z), which is a 
solution of the Schrodinger wave equation. The introduction 
of the spin wave functions for systems containing several electrons 
will be discussed later. 

Various other simplified methods of treating electron spin have 
been developed, such as those of Pauli, 1 Darwin, 2 and Dirac. 3 
These are especially useful in the approximate evaluation of 
interaction energies involving electron spins in systems containing 
more than one electron. 

29. THE HELIUM ATOM. THE PAULI EXCLUSION PRINCIPLE 

29a. The Configurations 1525 and 152^. — In Section 236 we 
applied the first-order perturbation theory to the normal helium 
atom. Let us now similarly discuss the first excited states of 
this atom, 4 arising from the unperturbed level for which one 

1 W. Pauli, Z. f. Phya. 43, 601 (1927). 

* C. G. Darwin, Proc. Roy. Soc. A116, 227 (1927). 

8 P. A. M. Dirac, Proc. Roy. Soc. A123, 714 (1929). 

4 This was first done by W. Heisenberg, Z. f. Phys. 39, 499 (1926). 



Vm-29a] 



THE HELIUM ATOM 



211 



electron has the total quantum number n = 1 and the other 
n = 2. It was shown that, if the interelectronic interaction 
term e 2 /ri 2 be considered as a perturbation, the solutions of the 
unperturbed wave equation are the products of two hydrogen- 
like wave functions 

in which the symbol (1) represents the coordinates (r u th, <pi) 
of the first electron, and (2) those of the second electron. The 
corresponding zeroth-order energy is 



W° nn = -^Rhc(- 2 + -\ 



We shall ignore the contribution of electron spin to the wave 
function until the next section. 

The first excited level, with the energy W Q = —5Rhc, is that 
for m = 1, n 2 = 2 and rii = 2, n 2 = 1. This is eight-fold 
degenerate, the eight corresponding zeroth-order wave functions 
being 



i«(D 

2»(1) 

1«(1) 

2p,(l) 

1.(1) 

2p„(l) 

1.(1) 

2p.(l) 



2s(2), 
1«(2), 
2p,(2)J 
1«(2), 

2p»(2),/ 

1«(2), 

2p.(2),] 

18(2), 



(29-1) 



in which we have chosen to use the real y functions and have 
represented ^ioo(l) by ls(l), and so on. 

On setting up the secular equation, it is found to have the 
form 



j, - w 



K. 



K. 


/. - w 






































J p - 


w 




Kp 




























K P 




Jp 




w 






































Jp - w 




Kp 






























K P 


Jp 


- w 






































Jp 


- w 




Kp 






























Kp 


Jp 


- W 



(2&~2) 



212 THE SPINNING ELECTRON [VIII-29a 

Here the symbols J», K a , J vy and K v represent the perturbation 
integrals 






(29-3) 



1«(1) Sto(2) — 1«(1) 25(2) dudTz, 

7*12 

1«(1) 2s(2) — 2s(l) 1«(2) dndr h 
1*(1) 2p,(2) -^ 1«(1) 2p x (2) dndrt, | 

#p = f fl*(l) 2px(2) —- 2p,(l) 1*(2) dndr,. 

J p and 2£ p also represent the integrals obtained by replacing 

2p x by 2p„ or 2p*, inasmuch as these three functions differ from 

one another only with regard to orientation in space. The 

integrals J, and J p are usually called Coulomb integrals; J,, 

for example, may be considered to represent the average Coulomb 

interaction energy of two electrons whose probability distribution 

functions are (ls(l)} 2 and {2s(2)} 2 . The integrals K 9 and K p 

are usually called resonance integrals (Sec. 41), and sometimes 

exchange integrals or interchange integrals, since the two wave 

functions involved differ from one another in the interchange 

of the electrons. 

It can be seen from symmetry arguments that all the remaining 

perturbation integrals vanish; we shall discuss /Jls(l) 2s(2) 

e 2 

— ls(l) 2p*(2) dridri as an example. In this integral the func- 

^12 

tion 2p x (2) is an odd function of the coordinate x 2 , and inasmuch 
as all the other terms in the integrand are even functions of a* 2 , 
the integral will vanish, the contribution from a region with x 2 
negative canceling that from the corresponding region with x 2 
positive. 

The solution of Equation 29-2 leads to the perturbation energy 
values 

W = J. + K., 

J v + K Pf (triple root), ( (2 *^ ) 

J p — K p , (triple root).j 

The splitting of the unperturbed level represented by these 
equations is shown in Figure 29-1. 



Vin-29a] 



THE HELIUM ATOM 



213 



One part of the splitting, due to the difference of the Coulomb 
integrals J 9 and J p , can be easily interpreted as resulting from the 
difference in the interaction of an inner Is electron with an 
outer 2s electron or 2p electron. This effect was recognized in 
the days of the old quantum theory, being described as resulting 
from greater penetration of the core of the atom (the nucleus 
plus the inner electrons) by the more eccentric orbits of the 



W 



ni,ryl,e 





Fig. 29-1. — The splitting of energy levels for the helium atom. 

outer electron, with a consequent increase in stability, an s orbit 
being more stable than a p orbit with the same value of n, and 
so on. 1 (It is this dependence of the energy of an electron on I 
as well as n which causes the energy levels of an atom to depend 
largely on the electronic configuration, this expression meaning 
the n and I values of all electrons. These values are usually 
indicated by writing ns, np 7 etc., with the number of similar 
electrons indicated by a superscript. Thus Is 2 indicates two Is 
electrons, ls 2 2p these plus a 2p electron, and so on.) 



1 Pauling and Goudsmit, "The Structure of Line Spectra," Chap. Ill 



214 THE SPINNING ELECTRON [VIH-29b 

On the other hand, the further splitting due to the integrals 
K, and K p was not satisfactorily interpreted before the develop- 
ment of the quantum mechanics. It will be shown in Section 41 
that we may describe it as resulting from the resonance phe- 
nomenon of the quantum mechanics. The zeroth-order wave 
function for the state with W = J 8 + K s , for example, is 

-i {1«(1)2*(2)+ 2 5 (1)1 5 (2)1; 

the atom in this state may be described as resonating between the 
structure in which the first electron is in the Is orbit and the 
second in the 2s orbit and that in which the electrons have been 
interchanged. 

A wave function of the type just mentioned is said to be 
symmetric in the positional coordinates of the two electrons, inasmuch 
as the interchange of the coordinates of the two electrons leaves 
the function unchanged. On the other hand, the wave function 

-~^{ls(l)2s(2) -2«(1) ls(2)} 

is antisymmetric in the positional coordinates of the electrons, 
their interchange causing the function to change sign. It is 
found that all wave functions for a system containing two 
identical particles are either symmetric or antisymmetric in the 
coordinates of the particles. 

For reasons discussed in the next section, the stationary states 
of two-electron atoms represented by symmetric and by anti- 
symmetric positional wave functions are called singlet states 
and triplet states, respectively. The triplet state from a given 
configuration is in general more stable than the singlet state. 

29b. The Consideration of Electron Spin. The Pauli Exclu- 
sion Principle. — In reconsidering' the above perturbation prob- 
lem, taking cognizance of the spin of the electrons, we must 
deal with thirty-two initial spin-orbit wave functions instead of 
the eight orbital functions ls(l) 2s(2), ls(l) 2p*(2), etc. These 
thirty-two functions are obtained by multiplying each of the 
eight orbital functions by each one of the four spin functions 

a(l) a(2), 

<*(D 0(2), 

0(1) «(2), 

0(1) 0(2). 



Vin-29b] 



THE HELIUM ATOM 



215 



Instead of using the second and third of these, it is convenient to 
use certain linear combinations of them, taking as the four spin 
functions for two electrons 



«(1) «(2), 
-^={a(l)0(2)+/9(l)a(2)} f j 

0(1) 0(2), 
i«(l)l8(2) -0(1)«(2)| 



(2&-5) 



1 

V2 



These are normalized and mutually orthogonal. The first 
three of them are symmetric in the spin coordinates of the two 
electrons, and the fourth is antisymmetric. It can be shown that 
these are correct zeroth-order spin functions for a perturbation 
involving the spins of the two electrons. 

Taking the thirty-two orbit functions in the order 

ls(l) 28(2) a(l) a(2), 
28(1) 18(2) o(l) o(2), 
18(1) 2p x (2) «(1) a(2), 



1«(1) 28(2) • 7^{«(D /?(2) + /?(!) a(2)}, 



obtained by multiplying the eight orbital functions by the first 
spin function, then by the second spin function, and so on, 
we find that the secular equation has the form 






































= 0, 



216 THE SPINNING ELECTRON [Vni-29b 

in which each of the small squares is an eight-rowed determinant 
identical with that of Equation 29-2. The integrals outside of 
these squares vanish because of the orthogonality of the spin 
functions and the non-occurrence of the spin coordinates in the 
perturbation function e 2 /r X2 . The roots of this equation are 
the same as those of Equation 29-2, each occurring four times, 
however, because of the four spin functions. 

The correct zeroth-order wave functions are obtained by 
multiplying the correct positional wave functions obtained in the 
preceding section by the four spin functions. For the con- 
figuration ls2s alone they are 

-L{1*(1) 2s(2) + 2 5 (1) 1«(2)} • «(1) a(2), 
7^{1«(1) 2s(2) + 2«(1) 1«(2)} • 7^{«(1) /8(2) + 0(1) «(2)} 
;^§{1«(1) 2»(2) + 2^(1) 1«(2)} • 0(1) 0(2), 
4={1«(1) 25(2) - 25(1) 15(2)} • 4={«(1) 0(2) ~ 0(1) «(2)}, 
f~L{l5(l) 25(2) - 25(1) 15(2)} . «(1) a(2), 

Triplet^ 1 *^ 2S(2) - 2S(1) 1S(2)} ' ^ { ^ (1) ^ (2) + 



1 {15(1)25(2) -2s(l)l5(2)}-0(l)0(2), 



0(1) «(2)} 



^V2 



Singlet 7^{1«(1) &(2) + 25(1) ls(2)} • 4={«(1) 0(2) - 

0(l)a(2)|. 

Of these eight functions, the first four are symmetric in the 
coordinates of the two electrons, the functions being unchanged 
on interchanging these coordinates. This symmetric character 
results for the first three functions from the symmetric character 
of the orbital part and of the spin part of each function. For 
the fourth function it results from the antisymmetric character 
of the two parts of the function, each of which changes sign on 
interchanging the two electrons. 



Vin-29b] THE HELIUM ATOM 217 

The remaining four functions are antisymmetric in the two 
electrons, either the orbital part being antisymmetric and the 
spin part symmetric, or the orbital part symmetric and the spin 
part antisymmetric. 

Just as for \s2s, so each configuration leads to some symmetric 
and some antisymmetric wave functions. For Is 2 , for example, 
there are three of the former and one of the latter, obtained by 
combining the symmetric orbital wave function of Section 236 
with the four spin functions. For ls2p there are twelve of each 



ls?p 



Symmetric Antisymmetric 

• • • 

• • • 



• • • 

• •• 01s2p'P 



• • • 

O O O • • • ls2p 3 P 

• •• 



O ls2» l S 
• • #ls2s*S 




Is 2 • • • Ols 2 'S 

Fig. 29-2. — Levels for configurations la 1 , ls2a, and la2p of the helium atom. $, 
spin-symmetric wave functions; Ot spin-antisymmetric wave functions 

type, nine spin-symmetric and orbital-symmetric, three spin- 
antisymmetric and orbital-antisymmetric; nine spin-svmmetric 
and orbital-antisymmetric, and three spin-antisymmetric and 
orbital-symmetric. The levels thus obtained for the helium 
atom by solution of the wave equation are shown in Figure 29-2, 
the completely symmetric wave functions being represented 
on the left and the completely antisymmetric ones on the right. 
Now it can be shown that if a helium atom is initially in a 
symmetric state no perturbation whatever will suffice to cause 
it to change to any except symmetric states (the two electrons 
being considered to be identical). Similarly, if it is initially 
in an antisymmetric state it will remain in an antisymmetric 
state. The solution of the wave equation has provided us with 



218 THE SPINNING ELECTRON [VIII-29b 

two completely independent sets of wave functions. To show 
that no perturbation will cause the system in a state represented 
by the symmetric wave function yp s to change to a state repre- 
sented by the antisymmetric wave function ^a we need only 
show that the integral 

vanishes (R' being the perturbation function, involving the spin 
as well as the positional coordinates of the electrons), inasmuch 
as it is shown in Chapter XI that the probability of transition 
from one stationary state to another as a result of a perturbation 
is determined by this integral. Now, if the electrons are identi- 
cal, the expression H'fa is a symmetrical function of the coordi- 
nates, whereas \p% is antisymmetric; hence the integrand will 
change sign on interchanging the coordinates of the two electrons, 
and since the region of integration is symmetrical in these 
coordinates, the contribution of one element of configuration 
space is just balanced by that of the element corresponding 
to the interchange of the electrons, and the integral vanishes. 1 

The question as to which types of wave functions actually 
occur in nature can at present be answered only by recourse to 
experiment. So far all observations which have been made on 
helium atoms have shown them to be in antisymmetric states. 2 
We accordingly make the additional postulate that the wave 
function representing an actual state of a system containing two 
or more electrons must be completely antisymmetric in the coordinates 
of the electrons; that is, on interchanging the coordinates of any 
two electrons it must change its sign. This is the statement of 
the Pauli exclusion principle in wave-mechanical language. 

This is a principle of the greatest importance. A universe 
based on some other principle, that is, represented by wave 
functions of different symmetry character, would be completely 
different in nature from our own universe. The chemical 
properties in particular of substances are determined by this 
principle, which, for example, restricts the population of the 
K shell of an atom to two electrons, and thus makes lithium 

1 The same conclusion is reached from the following argument: On inter- 
changing the subscripts 1 and 2 the entire integral is converted into itself 
with the negative sign, and hence its value must be zero. 

2 The states are identified through the splitting due to spin-orbit inter- 
actions neglected in our treatment. 



Vin-29b] THE HELIUM ATOM 219 

an alkali metal, the third electron being forced into an outer 
shell where it is only loosely bound. 

To show this, we may mention that if A represents a spin-orbit 
function for one electron (such that A(l) = ls(l) a(l), for 
example) and B, C, • • • , E others, then the determinantal 
function 

A{\) B(l) . . . E(\) 
A{2) 5(2) . . . E(2) 

A(N) B(N) . . . E(N) 

is completely antisymmetric in the N electrons, and hence a 
wave function of this form for the iV-electron system satisfies 
Paulas principle, since from the properties of determinants the 
interchange of two rows changes the sign of the determinant. 
Moreover, no two of the functions A, B } • • • , E can be equal, 
as then the determinant would vanish. Since the only spin-orbit 
functions based on a given one-electron orbital function are the 
two obtained by multiplying by the two spin functions a and 0, 
we see that no more than two electrons can occupy the same orbital 
in an atom, and these two must have their spins opposed; in other 
words, no two electrons in an atom can have the same values of 
the four quantum numbers w, I, ra, and m 8 . Paulas original 
statement 1 of his exclusion principle was in nearly this language; 
its name is due to its limitation of the number of electrons in an 
orbit. 

The equations of quantum statistical mechanics for a system of 
non-identical particles, for which all solutions of the wave 
equations are accepted, are closely analogous to the equations 
of classical statistical mechanics (Boltzmann statistics). The 
quantum statistics resulting from the acceptance of only anti- 
symmetric wave functions is considerably different. This 
statistics, called Fermi-Dirac statistics, applies to many problems, 
such as the Pauli-Sommerfeld treatment of metallic electrons 
and the Thomas-Fermi treatment of many-electron atoms. 
The statistics corresponding to the acceptance of only the 
completely symmetric wave functions is called the Bose-Einstein 
statistics. These statistics will be briefly discussed in Section 49. 

1 W. Pauli, Z. f. Phys. 31, 765 (1925). 



220 THE SPINNING ELECTRON [Vm-29b 

It has been found that for protons as well as electrons the wave 
functions representing states occurring in nature are antisym- 
metric in the coordinates of the particles, whereas for deuterons 
they are symmetric (Sec. 43/). 

The stationary states of the helium atom, represented on the 
right side of Figure 29-2, are conveniently divided into two sets, 
shown by open and closed circles, respectively. The wave 
functions for the former, called singlet states, are obtained by 
multiplying the symmetric orbital wave functions by the single 

antisymmetric spin function — j=.{a(l) 0(2) — 0(1) «(2)}. 

V2 

Those for the latter, called triplet states, are obtained by mul- 
tiplying the antisymmetric orbital wave functions by the three 
symmetric spin functions. 1 The spin-orbit interactions which 
we have neglected cause some of the triplet levels to be split 
into three adjacent levels. Transitions from a triplet to a 
singlet level can result only from a perturbation involving the 
electron spins, and since interaction of electron spins is small 
for light atoms, these transitions are infrequent; no spectral line 
resulting from such a transition has been observed for helium. 

It is customary to represent the spectral state of an atom by 
a term symbol such as l S, Z S, Z P, etc. Here the superscript on 
the left gives the multiplicity, 1 signifying singlet and 3 triplet. 
The letters S, P, etc., represent the resultant of the orbital 
angular-momentum vectors of all the electrons in the atom. 
This is also given by a resultant azimuthal quantum number L, 
the symbols S,P,D,F, • • • corresponding to L = 0,1,2,3, • * • . 
If all the electrons but one occupy s orbitals, the value of L is 
the same as that of I for the odd electron, so that for helium the 
configurations Is 2 , ls2s, and ls2p Jead to the states l S, l S and 3 >S, 
and l P and 3 P. Use is also made of a resultant spin quantum 
number S (not to be confused with the symbol S for L = 0), 

1 The electrons are often said to have their spins opposed or antiparallel 
in singlet states and parallel in triplet states, the spin function 

-^{« (1)0(2) +0(1) «(2)} 

in the latter case representing orientation of the resultant spin with zero 
component along the z axis. 



Vm-29c] THE HELIUM ATOM 221 

which has the value for singlet states and 1 for triplet states, 
the multiplicity being equal to 2S + l. 1 

The results which we have obtained regarding the stationary 
states of two-electron atoms may be summarized in the following 
way. The main factors determining the term values are the 
values of the principal quantum numbers fti and n 2 for the 
two electrons and of the azimuthal quantum numbers h and h t 
smaller values of these numbers leading to greater stability. 
These numbers determine the configuration of the atom. The 
configuration Is 2 leads to the normal state, ls2s to the next most 
stable states, then ls2p, and so on. For configurations with 
n\l\ different from n 2 ?2 there is a further splitting of the levels 
for a given configuration, due to the resonance integrals, 
leading to singlet and triplet levels, and to levels with different 
values of the resultant azimuthal quantum number L in case 
that both h and Z 2 are greater than zero. The triplet levels may 
be further split into their fine-structure components by the 
spin-orbit interaction, which we have neglected in our treatment. 
It is interesting to notice that these interactions completely 
remove the degeneracy for some states, such as ls2s l S, 
but not for others, such as ls2s Z S, which then show a further 
splitting (Zeeman effect) on the application of a magnetic field 
to the atom. 

Problem 29-1. Evaluate the integrals J and K for ls2s and ls2p of 
helium, and calculate by the first-order perturbation theory the term values 
for the levels obtained from these configurations. Observed term values 
(relative to He + ) are ls2s l S 32033, U2s *S 38455, ls2p l P 27176, and l«2p 3 P 
29233 cm" 1 . 

29c. The Accurate Treatment of the Normal Helium Atom. — 

The theoretical calculation of the energy of the normal helium 
atom proved to be an effective stumbling block for the old 
quantum theory. On the other hand, we have already seen that 
even the first attack on the problem by wave-mechanical methods, 

1 For a detailed discussion of spectroscopic nomenclature and the vector 
model of the atom see Pauling and Goudsmit. "The Structure of Line 
Spectra." The triplet levels of helium were long called doublets, complete 
resolution being difficult. Their triplet character was first suggested by 
J. C. Slater, Proc. Nat. Acad. Set. 11, 732 (1925), and was soon verified 
experimentally by W. V. Houston, Phys. Rev. 29, 749 (1927). The names 
parhelium and orthohelium were ascribed to the singlet and triplet levels, 
respectively, before their nature was understood. 



222 THE SPINNING ELECTRON [Vm-29c 

the first-order perturbation treatment given in Section 236, 
led to a promising result, the discrepancy of about 4 v.e. (accept- 
ing the experimental value as correct) being small compared 
with the discrepancies shown by the old-quantum-theory 
calculations. It is of interest to see whether or not more and 
more refined wave-mechanical treatments continue to diminish 
the discrepancy with experiment and ultimately to provide a 
theoretical value of the ionization potential agreeing exactly 
with the experimental (spectroscopic) value 24.463 v. 1 The 
success of this program would strengthen our confidence in our 
wave-mechanical equations, and permit us to proceed to the 
discussion of many-electron atoms and molecules. 

No exact solution of the wave equation has been made, and all 
investigators have used the variation method. 2 The simplest 
extension of the zeroth-order wave function e~ 2 *, with s = (ri + r 2 )/ 
a , is to introduce an effective nuclear charge Z'e in place of the 
true nuclear charge 2e in the wave function. This function, e~ z ' 9 , 
minimizes the energy when the atomic number Z' has the value 
2 K6> corresponding to a screening constant of value %$ (Sec. 
266). The discrepancy with the observed energy 3 (Table 29-1) 
is reduced by this simple change to 1.5 v.e., which is one-third the 
discrepancy for UnsolcTs treatment. This wave function cor- 
responds to assuming that each electron screens the other 

1 Calculated from Lyman's term value 198298 cm"" 1 corrected by Paschen 
to 198307.9 cm"" 1 ; T. Lyman, Astrophys. J. 60, 1 (1924); F. Paschen, Sitzber. 
preuss. Akad. Wiss. 1929, p. 662. 

2 The principal papers dealing with the normal helium atom are A. Unsold, 
Ann. d. Phys. 82, 355 (1927); G. W. Kellner, Z.f. Phys. 44, 91, 110 (1927); 
J. C. Slater, Proc. Nat. Acad. Set. 13, 423 (1927); Phys. Rev. 32, 349 (1928); 
C. Eckart, Phys. Rev. 36, 878 (1930); E. A. Hylleraas, Z. f. Phys. 48, 469 
(1928); 54, 347 (1929); 65, 209 (1930). A summary of his work is given by 
Hylleraas in Skrifter det Nor she Vid.-Ak. Oslo, I. Mat.-Naturv. Klasse 1932, 
pp. 5-141. For the special methods of evaluating and minimizing the 
energy integral, the reader is referred to these papers. 

3 The experimental value —78.605 v.e. = —5.8074 R^hc for the energy 
of the normal helium atom is obtained by adding to the observed first 
ionization energy 24.463 v,e. (with the minus sign) the energy 

4R Kt hc - -54.1416 v.e. 

of the helium ion. Hylleraas has shown that the correction for motion of 
the nucleus in the neutral helium atom is to be made approximately by 
using i^HeJ that is, by assigning to each electron the reduced mass with the 
helium nucleus. 



Vin-29c] THE HELIUM ATOM 223 

from the nucleus in the same way as a charge —%ee on the 
nucleus. 

Problem 29-2. (a) Calculate approximately the energy the normal 
lithium atom would have if the allowed wave functions were completely 
symmetric in the electrons, using for the positional wave function the 
product function ls(l) ls(2) ls(3), in which Is contains the effective nuclear 
charge Z' = 3 — S, and minimizing the energy relative to Z' or S. From 
this and a similar treatment of Li+ obtain the first ionization energy. The 
observed value is 5.368 v.e. (b) Obtain a general formula for the Nth 
ionization energy of an atom with atomic number Z in such a Bose-Einstein 
universe, using screening-constant wave functions. Note the absence of 
periodicity in the dependence on Z. 

We might now consider other functions of the type F(ri)F(r 2 ) y 
introducing other parameters. This has been done in a general 
way by Hartree, in applying his theory of the self-consistent 
field (Chap. IX), the function F(r x ) being evaluated by special 
numerical and graphical methods. The resulting energy value, 
as given in Table 29-1, is still 0.81 v.e. from the experimental 
value. Even the simple algebraic function 

Z\r\ Zyr\ Ztri Zyrt 

leads to as good a value of the energy. (This is function 4 of 
the table, there expressed in terms of the hyperbolic cosine.) 
This variation function we may interpret as representing one 
electron in an inner orbit and the other in an outer orbit, the 
values of the constants, Z x = 2.15 and Z 2 = 1.19, corresponding 
to no shielding (or, rather, a small negative shielding) for the 
inner electron by the outer, and nearly complete shielding for 
the outer electron by the inner. By taking the sum of two 
terms the orbital wave function is made symmetric in the two 
electrons. It is interesting that the still simpler function 
5 leads to a slightly better value for the energy. Various more 
complicated functions of s and t were also tried by Kellner and 
Hylleraas, with considerable improvement of the energy value. 
Then a major advance was made by Hylleraas by introducing 
in the wave function the coordinate u = rn/a 0t which occurs 
in the interaction term for the two electrons. The simple two- 
parameter functions 6 and 7 provide values of the energy of the 
atom accurate to H P er cen ^. Here again the polynomial in u 
is more satisfactory than the more complicated exponential 



224 



THE SPINNING ELECTRON 



[VIII-29C 



function, suggesting that a polynomial factor containing further 
powers of u } t, and s be used. The functions 8, 9, and 10 show 
that this procedure leads quickly to a value which is only slightly 
changed by further terms, the last three terms of 10 being 
reported by Hylleraas as making negligible contributions. 
The final theoretical value for the energy of the helium atom is 
0.0016 v.e. below the experimental value. Inasmuch as this 
theoretical value, obtained by the variation method, should 
be an upper limit, the discrepancy is to be attributed to a numer- 
ical error in the calculations or to experimental error in the 
ionization energy, or possibly to some small effects such as 
electron-spin interactions, motion of the nucleus, etc. At 
any rate the agreement to within 0.0016 v.e. may be considered 
as a triumph for wave mechanics when applied to many-electron 
atoms. 



Table 29-1. — Variation Functions for the Normal Helium Atom 1 



Symbols: s = » t = — 

do 
Experimental value of W = -5.80736/2 H «^c 



— r 2 r 12 

1 u = — 

ao cio 



Variation function, with best values of 
constants 2 



1. 6" 2 * 

2. e~ z '% Z' = 27 / 16 = 1.6875 

3. F(r x )F(r 2 ) 

4. «-*'• cosh ct, Z' = t.67, c = 0,48. . . 

5. e~ z, '(l + c 2 < 2 ), Z' = 1.69, c 2 = 0.142. . . 

6. e-*V u , Z' = 1.86, c = 0.26 

7. e~ z '»(l + citi), Z' = 1.849, c, = 0.364. . 

8. e~ zf '(l -h cxu 4- c 2 t 2 ) 

Z' = 1.816, c, = 0.30, c 2 = 0.13. 

9. e~ z, '{\ + ciu + c 2 t 2 + c 3 s + c 4 s 2 -f c 6 w 2 ) 
Z' = 1.818, ci = 0.353, c 2 = 0.128, 

c, - -0.101, ca = 0.033, c 6 = -0.032 
10. e~ z '* (polynomial with fourteen terms) . . 



Energy, 
in units 
— RnJic 



50 

6953 

75 

7508 
7536 
7792 

7824 
80488 



5.80648 



5.80748 



Difference with 
experiment 



Units 
—Rjuhc 



0.31 

0.1120 

0.06 

0.0565 

0.0537 

0.0281 

0.0249 

0.00245 

0.00085 



-0.00012 



V.e. 



4.19 

1.53 

0.81 

0.764 

0.726 

0.380 

0.337 

0.0332 

0115 



-0.0016 



1 A few variation functions which have been tried are not included in the table because 
they are only slightly better than simpler ones; for example, the function e~*'* (1 — ae~ c n u ), 
which is scarcely better than function 6. (D. R. Hartree and A. L. Ingman, Mem. 
Manchester Phil. Soc. 77, 69 (1932).) 

5 The normalization factor is omitted. Of these functions, 1 is due to Unsold, 2 to Kell- 
ner, 3 to Hartree and Gaunt, 4 to Eckart and Hylleraas, and the remainder to Hylleraas. 



VIII-29d] 



THE HELIUM ATOM 



225 



Hylleraas's masterly attack on the problem of the energy of 
normal helium and heliumlike ions culminated in his derivation 
of a general formula for the first ionization energy J of these 
atoms and ions. 1 This formula, obtained by purely theoretical 
considerations, is 



/ = 



RJic 



1 + 



— (z> - \l 

m Q y 4 

M 



Z + 0.31488 



0.01752 0.00548 \ 

z + z* )' 



(29-6) 



in which M is the mass and Z the atomic number of the atom. 
Values calculated by this formula 2 are given in Table 29-2, 
together with experimental values obtained spectroscopically, 
mainly by Edl£n 3 and coworkers. It is seen that there is agree- 

Table 29-2. — Ionization Energies of Two-electron Atoms 



Atom 


/ calculated, v.e. 


/ observed, v.e. 


H- 


0.7149 




He 


24.465 


24.463 


Li+ 


75.257 


75.279 ± 0.012 


Be + + 


153.109 


153.09 ±0.10 


B+++ 


258.029 


258.1 ±0.2 


C 4 + 


390.020 


389.9 ±0.4 


N 5 + 


549.085 




Q6+ 


735.222 





ment to within the experimental error. Indeed, the calculated 
values are now accepted as reliable by spectroscopists. 4 

Included in the table is the value 0.7149 v.e. for the ionization 
energy of the negative hydrogen ion H~~. This shows that the 
hydrogen atom has a positive electron affinity, ^amounting to 
16480 cal/mole. The consideration of the crystal energy of the 
alkali hydrides has provided a rough verification of this value. 

29d. Excited States of the Helium Atom. — The variation 
method can be applied to the lowest triplet state of helium as 
well as to the lowest singlet state, inasmuch as (neglecting 

iE. A. Hylleraas, Z.f. Phys. 66, 209 (1930). 

2 Using 1 v.e. = 8106.31 cm" 1 and R* = 109737.42 cm" 1 . 

3 A. Ericson and B. Edl6n, Nature 124, 688 (1929); Z. f. Phys. 69, 666 
(1930); B. Edl£n, Nature 127, 405 (1930). 

* B. Edl£n, Z. f. Phys. 84, 746 (1933). 



226 THE SPINNING ELECTRON [VIII-29e 

spin-orbit interactions) the triplet wave functions are anti- 
symmetric in the positional coordinates of the two electrons, 
and contain no contribution from singlet functions (Sec. 26c). 
A simple and reasonable variation function is 

1«*'(1) 2s*"(2) - 2sz-(l) lsz<(2), 

in which ls Z ' and 2s Z " signify hydrogenlike wave functions with 
the indicated effective nuclear charges as parameters. We 
would expect the energy to be minimized for Z r = 2 and Z" = 1. 
Calculations for this function have not been made. However, 
Hylleraas 1 has discussed the function 

se- z ' a sinh ct, (29-7) 

obtaining the energy value — 4.342(XR He Ac, not far above the 
observed value — 4.3504 J R He /ic. This function is similar to the 
hydrogenlike function (containing some additional terms), 
and the parameter values found, Z' = 1.374 and c = 0.825, 
correspond to the reasonable values Z' = 2.198 and Z" = 1.099. 
Hylleraas has also replaced s in 29-7 by s + CiU, obtaining the 
energy — 4.3448# H ^c, and by s + c 2 £ 2 , obtaining the energy 
— 4.3484#H e Ac. It is probable that the series s + CiU + c 2 t 2 
would lead to very close agreement with experiment. 

Numerous investigations by Hylleraas and others 2 have 
shown that wave mechanics can be applied in the treatment of 
other states of the helium atom. We shall not discuss further 
the rather complicated calculations. 

29e. The Polarizability of the Normal Helium Atom.— A 
quantity of importance for many physical and chemical con- 
siderations (indices of refraction, electric dipole moments, 
term values of non-penetrating orbits, van der Waals forces, 
etc.) is the polarizability of atoms and molecules, mentioned in 
Problem 26-1 and Sections 27a and 27e. We may write as the 
energy of a system in an electric field of strength F the expression 

W = W° - %aF* + • • • (29-8) 

1 E. A. Hylleraas, Z. j. Phys. 64, 347 (1929). 

2 W. Heisenberg, Z. f. Phys. 39, 499 (1926); A. Unsold, Ann. d. Phys. 
82, 355 (1927); E. A. Hylleraas and B. Undheim, Z.f. Phys. 66, 759 (1930); 
E. A. Hylleraas, ibid. 66, 453 (1930); 83, 739 (1933); J. P. Smith, Phys. 
Rev. 42, 176 (1932); etc. 



VIII-29e] THE HELIUM ATOM 227 

in case that the term linear in F vanishes, the permanent electric 
moment of the system being zero. The electric moment induced 
in the system by the field is aF } the factor of proportionality a. 
being called the polarizability. The polarizability of the 
molecules in a gas determines its index of refraction n (for light 
of very large wave length) and its dielectric constant D, according 
to the equation 

N 3 n 2 -l 3 P- 1 , 9Q _ Q . 

V a =^tf + 2 = ±*D-+2' (29 ~ 9) 

in which N is Avogadro's number and V is the molal volume of 
the substance. The mole refraction R is defined as 

R = —-a = 2.54 • 10 24 <*. (29-10) 

The dimensions of R and a are those of volume, and their magni- 
tudes are roughly those of molal volumes and molecular volumes, 
respectively; for example, for monatomic hydrogen 22 = 1.69 cm 3 
and a = 0.667 • 10~ 24 cm 3 (Sec. 27a). Values of R and a are 
determined experimentally mainly by measurement of indices 
of refraction and of dielectric constants, 1 rough values being 
also obtainable from spectral data. 2 

The value of the polarizability a of an atom or molecule can 
be calculated by evaluating the second-order Stark effect energy 
— %aF 2 by the methods of perturbation theory or by other 
approximate methods. A discussion of the hydrogen atom has 
been given in Sections 27a and 27 e (and Problem 26-1). The 
helium atom has been treated by various investigators by the 
variation method, and an extensive approximate treatment 
of many-electron atoms and ions based on the use of screening 
constants (Sec. 33a) has also been given. 3 We shall discuss the 
variational treatments of the helium atom in detail. 

The additional term in the Hamiltonian due to the electric 

1 The total polarization of a gas may be due to polarization of the electrons 
in the gas molecules (for fixed nuclear positions), polarization of the nuclei 
(with change in the relative positions of the nuclei in the molecules), and 
orientation of molecules with permanent electric dipole moments. We are 
here discussing only the first of these mechanisms; the second is usually 
unimportant, and the third is treated briefly in Section 49/. 

1 See Pauling and Gotjdsmit, "The Structure of Line Spectra," Sec. 11. 

8 L. Paulino, Proc. Roy. Soc. A114, 181 (1927). 



228 



THE SPINNING ELECTRON 



[Vin-29e 



field (assumed to lie along the z axis) is eF{z x + z 2 ), z x and z 2 
being the z coordinates of the two electrons relative to the nucleus. 
The argument of Section 27c suggests that the variation function 
be of the form 

* = *°{1 + («i + *0/(*i, Vi, *u **, 2/2, 22)}, (29-11) 

in which xp is an approximate wave function for zero field. 
Variation functions of this form (or approximating it) have been 
discussed by Hass6, Atanasoff, and Slater and Kirkwood, 1 
whose results are given in Table 29-3. 

Table 29-3. — Variation Functions for the Calculation of the 

pol ariz ability of the normal helium atom 

Experimental value: a = 0.205 • 10"" 24 cm 3 

r\ + r 2 

s = ■ 



Variation function 


a 


References 1 


1. 


e-*''|l -f A(zi + z 2 )\ 


0.150-10~ 24 cm 3 


H 


2. 


e -z>*\i _|_ A(zie~ z " r i + z 2 e- z " r z)} 


.164 


SK 


3. 


(rirj)°- 1M e-^«{l + A(zie" z " r i + zi<r z " r ')\ 


.222 


SK 


4. 


e- z ''[l +A(zi+ z 2 ) + B{z l r l + z 2 r 2 )) .. . . 


.182 


H 


5. 


6 -z't(i -1-^4(2! -f 2 2 ) -f- terms to quartic)} 


.183 


H 


6. 


e- z '*(l +c lU )[l +A( 2i +z 2 ) + 








Bfari -f z 2 r 2 )) 


.201 


H 


7. 


e~ z '>\\ +au + c 2 t 2 +A(zi +22)} 


.127 


A 


8. 


e~ z '*\l + citt + c 2 * 2 + W + £a)(zi + z*) 








+ Cf(2l ~ *«)} 


.182 


A 


9. 


g-Z'« j 1 + citt + c 2 * 2 + c 8 s + c 4 s 2 + c 5 w 2 
-f (A -f B«)(«i + 22) + Ct( Zl - z 2 ) 








+ DU( ZI +Z2)| 


.194 


A 


10. 


<r*'*(l + Cl w + c 2 * 2 )(l +A(Zi+ z 2 ) 








+B(*,r, + * 2 r 2 )} 


.231 


H 


11. 


A non-algebraic function 


.210 


SK 









1 H — Hass6, A - Atanasoff, SK = Slater and Kirkwood. 

Of these functions, 1, 2, 4, and 5 are based on the simple 
screening-constant function 2 of Table 29-1; these give low 
values of a, the experimental value (from indices of refraction 
extrapolated to large wave length of light and from dielectric 

1 H. R. Hasse, Proc. Cambridge Phil. Soc. 26, 542 (1930), 27, 66 (1931); 
J. V. Atanasoff, Phys. Rev. 36> 1232 (1930); J. C. Slater and J. G. Kirk- 
wood, Phys. Rev. 37, 682 (1931). 



Vin-29e1 THE HELIUM ATOM 229 

constant) being about 0.205 • 10~ 24 cm 3 . The third function, 
supposed to provide a better approximation to the correct wave 
function for large values of r\ and r 2 (that is, in the region of the 
atom in which most of the polarization presumably occurs), 
overshoots the mark somewhat. (The fundamental theorem 
of the variation method (Sec. 26a) does not require that a 
calculation such as these give a lower limit for a, inasmuch as 
the wave function and energy value for the unperturbed system 
as well as for the perturbed system are only approximate.) 
Function 6 is based on 7 of Table 29-1, 7, 8, and 10 on 8, and 
9 on 9. It is seen that functions of the form 29-11 (6, 10) seem 
to be somewhat superior to functions of the same complexity 
not of this form (7, 8, 9). Function 11 is based on a helium- 
atom function (not given by a single algebraic expression) due to 
Slater. 1 

It is seen that the values of a given by these calculations in 
the main lie within about 10 per cent of the experimental value 2 
0.205 • 10~ 24 cm 3 . For Li+, Hass6, using function 6, found the 
value a = 0.0313 • 10~ 24 cm 3 ; the only other values with which 
this can be compared are the spectroscopic value 3 0.025 and the 
screening-constant value 2 0.0291 • 10~ 24 cm 3 . 

Problem 29-3. Using the method of Section 27e and the screening- 
constant wave function 2 of Table 29-1, evaluate the polarizability of the 
helium atom, taking as the zero point for energy the singly ionized atom. 

1 J. C. Slater, Phys. Rev. 32, 349 (1928). 

2 The rough screening-constant treatment mentioned above gives the 
values 0.199 • 10" 24 cm 3 for He and 0.0291 • 10" 24 cm 3 for Li+. 

3 J. E. Mayer and M. G. -Mayer, PJiys. Rev. 43, 605 (1933). 



CHAPTER IX 
MANY-ELECTRON ATOMS 

Up to the present time no method has been applied to atoms 
with more than two electrons which makes possible the computa- 
tion of wave functions or energy levels as accurate as those for 
helium discussed in Section 29c. With the increasing complexity 
of the atom, the labor of making calculations similar to those 
used for the ground state of helium increases tremendously. 
Nevertheless, many calculations of an approximate nature have 
been carried out for larger atoms with results which have been 
of considerable value. We shall discuss some of these in this 
chapter. 1 

30. SLATER'S TREATMENT OF COMPLEX ATOMS 

30a. Exchange Degeneracy. — All of the methods which we 
shall consider are based on a first approximation in which the 
interaction of the electrons with each other has either been 
omitted or been replaced by a centrally symmetric field approxi- 
mately representing the average effect of all the other electrons 
on the one under consideration. We may first think of the prob- 
lem as a perturbation problem. The wave equation for an atom 
with N electrons and a stationary nucleus is 

*-l \ t' = l t,j>i I 

in which r, is the distance of the ith electron from the nucleus, 
Ta is the distance between the ith and jth electrons, and Z is 
the atomic number. 

If the terms in r# are omitted, this equation is separable into 
N three-dimensional equations, one for each electron, just as 
was found to be the case for helium in Section 236. To this 

1 This chapter can be omitted by readers not interested in atomic spectra 
and related subjects; however, the treatment is closely related to that for 
molecules given in Chapter XIII. 

230 



IX-30a] SLATER'S TREATMENT OF COMPLEX ATOMS 231 

degree of approximation the wave function for the atom may be 
built up out of single-electron wave functions; that is, a solution 
of the equation for the atom with 2e 2 /r# omitted is 

*! = Ua(l) u*(2) • • • u,(N), (30-2) 

in which t/«(l), etc., are the solutions of the separated single- 
electron equations with the three quantum numbers 1 symbolized 
by a, ($, • • • , v and the three coordinates symbolized by 
1, 2, • • • , N. With this form for ^° the individual electrons 
retain their identity and their own quantum numbers. How- 
ever, an equally good solution of the unperturbed equation cor- 
responding to the same energy as Equation 30-2 is 

+° = ua(2) UfiO) • • • u v (N) } (30-3) 

in which electrons 1 and 2 have been interchanged. In general, 
the function 

l# = Pu a (l) Ufi(2) • • • u v (N), (30-4) 

in which P is any permutation of the electron coordinates, is an 
unperturbed solution for this energy level. 

The meaning of the operator P may be illustrated by a simple example. 
Let us consider the permutations of the three symbols x i} z 2 , x%. These are 
Xi t Xi, Xz) x 2 , x%> x\) Xz, Xi, x 2 ; x 2 , xi, x z ; Xi, x z , x%\ x z , x 2 , x\. Any one of these 
six may be represented by Px i, x 2 , x Z) in which P represents the operation of 
permuting the symbols x h x 2 , x z in one of the above ways. The operation P 
which yields xi, x 2 , x z is called the identity operation. 

Any of the above permutations can be formed from x h x 2 , x z by successive 
interchanges of pairs of symbols. This can be done in more than one way, 
but the number of interchanges necessary is either always even or always 
odd, regardless of the manner in which it is carried out. A permutation 
is said to be even if it is equivalent to an even number of interchanges, and 
odd if it is equivalent to an odd number. We shall find it convenient to 
use the symbol ( — l) p to represent +1 when P is an even permutation and 
— 1 when P is an odd permutation. 

Multiplication of the operators P and P' means that P and P' are to be 
applied successively. The set of all the permutations of N symbols has the 
property that the product PP' of any two of them is equal to some other 
permutation of the set. A set of operators with this property is called 8 
group, if in addition the set possesses an identity operation and if every 
operation P possesses an inverse operation P" 1 such that PP" 1 is equivalent 
to the identity operation. There are #! permutations of N different 
symbols. 

1 The symbols a, ft • • • , v are of course not related to the spin functions 
a and 0. 



232 MANY-ELECTRON ATOMS [IX-30a 

At this point we may introduce the spin of the electrons into 
the wave function (in the same manner as for helium) by multi- 
plying each single-electron orbital function by either a(o>) or 
0(co). For convenience we shall include these spin factors in 
the functions u a (l), etc., so that hereafter a, j8, 7, • • • represent 
four quantum numbers n, I, mi, and m 8 for each electron and 1, 
2, • • • represent four coordinates r», & iy <p», and co». As discussed 
in Section 29a for the two-electron case, treatment of this 
degenerate energy level by perturbation theory (the electron 
interactions being the perturbation) leads to certain combinations 

r = 77Fi^ cpPUaW ^ (2) ' ' ' Uv(N) (3a ~ 5) 
p 

for the correct zeroth-order normalized wave functions. One 
of these combinations will have the value +1 for each of the 
coefficients c P . Interchange of any pair of electrons in this 
function leaves the function unchanged; i.e., it is completely 
symmetric in the electron coordinates. For another combina- 
tion the coefficients cp are equal to +1 or to — 1, according as P 
is an even or an odd permutation. This combination is com- 
pletely antisymmetric in the electrons; i.e., the interchange of 
any two electrons changes the sign of the function without 
otherwise altering it. Besides these two combinations, which 
were the only ones which occurred in helium, there are for 
many-electron atoms others which have intermediate symmetries. 
However, this complexity is entirely eliminated by the appli- 
cation of the Pauli exclusion principle (Sec. 29b) which says that 
only the completely antisymmetric combination 

r = ^2 (_1)i>Pwa(1) M " (2) ' ■ ■ u - {N) (30_6) 

p 

has physical significance. This solution may also be written as a 
determinant, 

U a (l) Ufi(l) . . . U v {\) 

u a {2) u (2) . . . tt„(2) 



^0 



VW~\ 



(30-7) 



Ua(N) u,(N) . . . u v (N) 
as was done in Section 296. The two forms are identical. 



IX-30b] SLATER'S TREATMENT OF COMPLEX ATOMS 233 

30b. Spatial Degeneracy. — In the previous section we have 

taken care of the degeneracy due to the N ! possible distributions; 
of the N electrons in a fixed set of N functions u. There still 
remains another type of degeneracy, due to the possibility of 
there being more than one set of spin-orbit functions correspond- 
ing to the same unperturbed energy. In particular there may 
be other sets of u's differing from the first in that one or more of 
the quantum numbers mi or m 8 have been changed. These 
quantum numbers, which represent the z components of orbital 
and spin angular momentum of the individual electrons, do not 
affect the unperturbed energy. It is therefore necessary for 
us to construct the secular equation for all these possible func- 
tions in order to find the correct combinations and first approxi- 
mation to the energy levels. 1 

Before doing this, however, we should ask if there are any more 
unperturbed wave functions belonging to this level. If, in setting 
up the perturbation problem, we had called the term 2e 2 /7\/ 
the perturbation, then the single-electron functions would have 
been hydrogenlike functions with quantum numbers n, l y mi, 
and m 8 . The energy of these solutions depends only on n, 
as we have seen. However, a better starting point is to add and 

subtract a term 2j)(xi) representing approximately the average 

i 

effect of the electrons on each other. If this term is added to 
H° and subtracted from H', the true Hamiltonian H = H° + H' 
is of course unaltered and the unperturbed equation is still 
separable. The single-electron functions are, however, no longer 
hydrogenlike functions and their energies are no longer inde- 
pendent of the quantum number I, because it is only with a 
Coulomb field that such a degeneracy exists (see Sec. 29a). 
Therefore, in considering the wave functions to be combined we 
do not ordinarily include any but those involving a single set of 
values of n and l\ i.e., those belonging to a single configuration. 
The consideration of a simple example, the configuration 
ls 2 2p of lithium, may make clearer what the different unper- 
turbed functions are. Table 30-1 gives the sets of quantum num- 

1 The treatment of atoms which we are giving is due to J. C. Slater, Phys. 
Rev. 34, 1293 (1929), who showed that this method was very much simpler 
and more powerful than the complicated group-theory methods previously 
used. 



234 



MANY-ELECTRON ATOMS 



[IXSOb 



bers possible for this configuration. The notation (1003^) 
means n = 1, I = 0, mi = 0, m, = +K- Each line of the table 
n corresponds to a set of functions u a • • • u v which when 
substituted into the determinant of Equation 30-7 gives a satis- 
factory antisymmetrical wave function ^J corresponding to the 

Table 30-1. — Sets of Quantum Numbers for the Configuration ls 2 2p 



1. (100K) (ioo -M) (2iiH); 

2. (100K) (100 -K) (211 -K); 

3. (lOOK) (ioo -H) (210K); 

4. (iooh) (ioo -K) (210 -H); 

5. (100K) (100 -K) (21 -IK); 

6. (iook) (ioo -K) (2i -l -K); 



2mi = +1, 2m, = +K, 

Smz = +1, 2m, = -K, 

Smj = 0, 2m, = +K> 

2w/ = 0, 2m, = -K, 

2m z = -1, 2m, = +K, 

2mj = — 1, 2m, = — K- 



same unperturbed energy level. No other sets satisfying the 
Pauli exclusion principle can be written for this configura- 
tion. The order of the expressions n, I, m t , m 8 in a given row is 
unimportant. 

This simple case illustrates the idea of completed shells of 
electrons. The first two sets of quantum numbers remain the 

Table 30-2. — Sets of Quantum Numbers for the Configuration np 2 





2mj 


2m, 


1. (nllK) 


(nil -K) 


2 





2. (nliy 2 ) 


(nlOK) 


1 


+ 1 


3. (nllK) 


(mo -y 2 ) 


1 





4. (nil ~K) 


(nlOK) 


1 





5. (nil ~K) 


(nlO -K) 


1 


-1 


6. (nllH) 


(nl -IK) 





+1 


7. (nliy 2 ) 


(nl-l-K) 








8. (nil ~K) 


(nl -IK) 








9. (nlOK) 


(nlO -K) 








10. (nil -K) 


(nl -1 -K) 





-1 


11. (nl -IK) 


(nlOK) 


-1 


+1 


12. (nl -1 -H) (nl0y 2 ) 


-1 





13. (nl -IK) 


(nlO -K) 


-1 





14. (nl -1 -K) (nlO -K) 


-1 


-1 


15. (nl -IK) 


(nl -l'-K) 


-2 






same throughout this table because Is 2 is a completed shell; 
i.e., it contains as many electrons as there are possible sets of 
quantum numbers. The shell ns can contain two electrons, 
np six electrons, nd ten electrons, etc. In determining the 



EX-30C] SLATER'S TREATMENT OF COMPLEX ATOMS 235 

number of wave functions which must be combined, it is only 
necessary to consider electrons outside of completed shells, 
because there can be only one set of functions Ua • • • u v i or the 
completed shells. 

Table 30-2 gives the allowed sets of quantum numbers for two 
equivalent p electrons, i.e., two electrons with the same value 
of n and with I = 1. 

Problem 30-1. Construct tables similar to Table 30-2 for the configura- 
tions np 3 and nd 2 . 

30c. Factorization and Solution of the Secular Equation. — We 

have now determined the unperturbed wave functions which 
must be combined in order to get the correct zeroth-order wave 
functions for the atom. The next step is to set up the secular 
equation for these functions as required by perturbation theory, 
the form given at the end of Section 24 being the most con- 
venient. This equation has the form 



#n - W H l2 • • • H lk 

#21 #22 — W * H 2 k 

Hki Hk2 ' ' ' Hkk — W 



= 0, (30-8) 



in which 



H nm = ftfH+mdr. (30-9) 



\p n is an antisymmetric normalized wave function of the form of 
Equation 30-6 or 30-7, the functions u composing it correspond- 
ing to the nth row of a table such as Table 30-1 or 30-2. H is 
the true Hamiltonian for the atom, including the interactions 
of the electrons. 

This equation is of the fcth degree, k being the number of 
allowed sets of functions u a • • • u v . Thus for the configura- 
tion ls 2 2p k is equal to 6, as is seen from Table 30-1. However, 
there is a theorem which greatly simplifies the solution of this 
equation: the integral H mn is zero unless \{/ m and \l/ n have the same 
value of Sm, and the same value of 2m i, these quantities being 
the sums of quantum numbers m 8 and mi of the functions u making 
up \p m and \l/ n . We shall prove this theorem in Section 30d in 
connection with the evaluation of the integrals H mn} and in the 
meantime we shall employ the result to factor the secular 
equation. 



236 



MANY-ELECTRON ATOMS 



[IX-30c 



Examining Table 30-1, we see that the secular equation for 
ls 2 2p factors into six linear factors; i.e., no two functions yj/ n 
and ^» have the same values of Sm 8 and Sm^. The equation 
for np 2 , as seen from Table 30-2, has the factors indicated by 
Figure 30-1, the shaded squares being the only non-zero elements. 
A fifteenth-degree equation has, therefore, by the use of this 
theorem been reduced to a cubic, two quadratic, and eight 
linear factors. 




Fig. 30-1. — The Becular determinant for the configuration np 2 , represented 

diagrammatically. 

By evaluating the integrals H mn and solving these equations, 
the approximate energy levels W corresponding to this con- 
figuration could be obtained; but a still simpler method is 
available, based on the fact that the roots W of the equations of 
lower degree will coincide with some of the roots of the equations 
of higher degree. The reason for this may be made clear by the 
following argument. The wave . functions \pi, ^ 2 , • • • , 4>kj 
which we are combining, differ from one another only in the 
quantum numbers m 8 and mi of the single electrons, these 
quantum numbers representing the z components of the spin 
and orbital angular momenta of the electrons. The energy 
of a single electron in a central field does not depend on mi or m 8 
(neglecting magnetic effects), since these quantum numbers 
refer essentially to orientation in space. The energy of an 
atom with several electrons does depend on these quantum 
numbers, because the mutual interaction of the 'electrons is 
influenced by the relative orientations of the angular-momentum 



IX-30c] SLATER'S TREATMENT OF COMPLEX ATOMS 237 

vectors of the individual electrons. Just as for one-electron 
atoms, however, the orientation of the whole atom in space 
does not affect its energy and we expect to find a number of states 
having the same energy but corresponding to different values 
of the z components of the total orbital angular momentum 
and of the total spin angular momentum; i.e., to different values 
of 2m i and 2m„ 

This type of argument is the basis of the vector model 1 for atoms, 
a very convenient method of illustrating and remembering the 
results of quantum-mechanical discussions such as the one we 
are giving here. In the vector model of the atom the orbital 
and spin angular momenta of the individual electrons are con- 
sidered as vectors (see Section le) which may be combined to 
give resultant vectors for the whole atom, the manner in which 
these vectors are allowed to combine being restricted by certain 
rules in such a way as to duplicate the results of quantum 
mechanics. The vector picture is especially useful in classifying 
and naming the energy levels of an atom, the values of the 
resultant vectors being used to specify the different levels. 

In Chapter XV we shall show that not only is the energy of a 
stationary state of a free atom a quantity which has a definite 
value (and not a probability distribution of values) but also the 
total angular momentum and the component of angular momen- 
tum in any one chosen direction (say the z direction) are similar 
quantities. Whereas it is not possible to specify exactly both the 
energy and the positions of the electrons in an atom, it is possible 
to specify the above three quantities simultaneously. If the 
magnetic effects are neglected we may go further and specify the 
total spin and total orbital angular momenta separately, and 
likewise their z components. However, we may not give 
the angular momenta of the individual electrons separately, 
these being quantities which fluctuate because of the electron 
interactions. 

It will likewise be shown that when magnetic effects are 
neglected the square of the total orbital angular momentum 
must assume only the quantized values L(L + l)(h/2w) 2 where 
L is an integer, while the square of the total spin angular momen- 
tum can take on only the values S(S + 1)(A/2t) 2 where S is 
integral or half-integral. (The letter L is usually used for the 

1 See Pauling and Gotjdsmit, "The Structure of Line Spectra." 



238 MANY-ELECTRON ATOMS [IX-30c 

total resultant orbital angular momentum of the atom, and the 
letter S for the total spin angular momentum; see Section 296.) 
In the approximation 1 we are using, the states of an atom may- 
be labeled by giving the configuration and the quantum numbers 
L, S, Ml = 2m*, and Ms = 2ra„ the last two having no effect 
on the energy. Just as for one electron, the allowed values 
of Ml are L, L — 1, • • • , — L + 1, — L] M s is similarly 
restricted to S, S — 1, • • • , —8 + 1, — S, all of these values 
of Ms and M L belonging to the same degenerate energy level and 
corresponding to different orientations in space of the vectors 
L and S. 

We shall now apply these ideas to the solution of the secular 
equation, taking the configuration np 2 as an example. From 
Table 30-2 we see that Hu — W is a linear factor of the equation, 
since ^i alone has 2mi = 2 and Xm 8 = 0. A state with Ml = 2 
must from the above considerations have L ^ 2. Since 2 is 
the highest value of Ml in the table, it must correspond to 
L = 2. Furthermore the state must have S = 0, because 
otherwise there would appear entries in the table with Ml = 2 
and Ms ^ 0. This same root W must appear five times in the 
secular equation, corresponding to the degenerate states L = 2, 
S = 0, M s = 0, Ml = 2, 1, 0, -1, -2. From this it is seen 
that this root (which can be obtained from the linear factors) 
must occur in two of the linear factors (Ml = 2,-2; Ms = 0), 
in two of the quadratic factors (Ml = 1,-1; Ms = 0), and in 
the cubic factor (Ml = 0, Ms = 0). The linear factor H 2 2 — W 
with Ml = 1, Ms = 1 must belong to the level L = 1, S = 1, 
because no terms with higher values of M L and Ms appear in 
the table except those already accounted for. This level will 
correspond to the nine states with Ml = 1,0, — 1, and Ms = 1, 
0, — 1. Six of these are roots of linear factors (M L = ±1, 
Ms = ±1; Ml = 0; M a = ±1), two of them are roots of the 
quadratic factors (Ml = ±1, Ms = 0), and one is a root of the 
cubic factor (M L = 0, Ms = 0). 

Without actually solving the quadratic equations or evaluating 
the integrals involved in them, we have determined their roots, 
since all the roots of the quadratics occur also in linear factors. 

1 This approximation, called (LS) or Russell-Saunders coupling, is valid 
for light atoms. Other approximations must be made for heavy atoms in 
which the magnetic effects are more important. 



EMOd] SLATER'S TREATMENT OF COMPLEX ATOMS 239 

Likewise we have obtained two of the three roots of the cubic. 
The third root of the cubic can be evaluated without solving the 
cubic or calculating the non-diagonal elements of the equation, 
by appealing to the theorem that the sum of the roots of a 
secular equation (or of one of its factors) is equal to the sum 
of the diagonal elements of the equation 1 (or of the factor). 
Since two of the roots of the cubic have been found and the sum 
of the roots is given by the theorem, the third may be found. 
It corresponds to the state L = 0, S = 0, since this is the only 
possibility left giving one state with Ml = and Ms = 0. 
The three energy levels for np 2 which we have found are 

W = # u , l D (L = 2, S = 0, M 8 - 0, Ml = 2, 1, 0, -1, -2);\ 
W = #22, 3 P (L = 1, S = 1, Ms = ±1, 0; Ml = ±1, 0);( 

W = #77 + #88 + #99 - #11 - #22, *S (L = 0, S = 0, 

Ms = 0, Ml = 0). 
(30-10) 

The term symbols 1 D, 3 P, *S have been explained in Section 296. 

Problem 30-2. Investigate the factorization of the secular equation 
for np 3 , using the results of Problem 30-1, and list terms which belong to 
this configuration. 

30d. Evaluation of Integrals. — We need to obtain expressions 
for integrals of the type 

# mn = j+*H+ n dr = ^22 ( "" l)p+p 'J p,w * (l) • • • 

p p' 
u*(N)HPu a (l) • • • u,(N)dr. (30-11) 

1 To prove this theorem, we expand the secular equation 30-8 and arrange 
according to powers of W. The resulting algebraic equation in W will have 
k roots, TTi, W2, • • • , Wk and can therefore be factored into k factors 

(W - WMW - W 2) • • • (W - W k ) = 0. 

The coefficient of — W k ~ l in this form of the equation is seen to be 

Wi + W 2 + • • • + W k ; 

the coefficient of - W k ~ l in Equation 30-8 is seen to be 

Hw + H22 + • • • ■+■ Hkk. 

These two expressions must therefore be equal, which proves the theorem. 



240 MANY-ELECTRON ATOMS [K-S0d 

We may eliminate one of the summations by the following 
device: 

JP f u*(l) • • • u*(N)HPu a (l) • • • u v {N)dr = 
P"JP'u*(l) • • • u*(N)HPu a (l) • • • !*,(#)*• = 

JP"P'u*(l) • • • u*(N)HP"Pu a (\) • • • !*,(#)*. 

the first step being allowed because P" only interchanges the 
names of the variables of a definite integral. If we choose P" 
to be P' -1 , the inverse permutation to P' 9 then P"P f = 1; i.e., 
P'T' is the identity operation, while P"P is still some member 
of the set of permutations, all members of which are summed 
over. The integral therefore no longer involves P f and the 
sum over P' reduces to multiplication by-iV!, the number of 
permutations. We thus obtain the equation 

H mn = 2(-l) p K*(D • • • u*(N)HPu a (l) • • - u v {N)dr. 
p 

(30-12) 

We shall now prove the theorem that H mn = unless 2m, is 
the same for ^ m and ^ n . i? does not involve the spin coordinates 
so that integration over these coordinates yields a product of 
orthogonality integrals for the spin functions of the various 
electrons. Unless the spins of corresponding electrons in the 
two functions u*(l) • • • u*(N) and Pu a (l) • • • u v (N) are 
the same, the integral is zero. If 2m, is not the same for \l/ m 
and ^ n there can be no permutation P which will make such a 
matching of the spins possible, because the number of positive 
and negative spins is different in the two functions. 

To prove the theorem concerning Sraj it is necessary to specify 
further the nature of H. We write 



H = %f t + Xf/a, 



«j>« 



where 



fi - -&^r„ v? - 77 and 9ii = ui 



The functions w«(l) • • • are solutions of 
( h 2 %e 2 ) 



EMOd] SLATER'S TREATMENT OF COMPLEX ATOMS 241 

From this we see that 

fiU&) =/(r<)w f (i), 

where /(r») is a function of r< alone. The integral of the first 
term in H thus reduces to 

X(-l) r %$<(\)Pu a {l)dn ■ ■ ■ fu?(i)f(ri)Pu t (i)dn ■ ■ • 

P I 

Su?{N)Pu v (N)dr N , (30-13) 

in which Pu^ (i) is used as a symbol for u r (j) in which electron j 
has replaced i as a result of the permutation P. Because of the 
orthogonality of the u'&, this is zero unless Pu^(i) = u^(j) except 
perhaps for j equal to the one value i. In addition, since 

uS) = Rm(Ti) • e lm (#i) • e im % (30-14) 

the factor fu?(i)f(ri)Pu{(i)dTi will be zero unless u f (i) and 
Puf(i) have the same quantum number m t . We thus see that 
this integral will vanish, unless all the u's but one pair match and 
the members of that pair have the same value of mi. 

Similar treatment of the term Xg^ shows that all but perhaps 
two pairs must match. The factor containing these unmatched 
functions is 

M(i)ut(j)fPut(i)Pui(j)dT4Ti. (30-15) 

It can be shown 1 that 

k -^5^4*^^*^^*'^^' (30_16) 

k,tn 

in which r a is the smaller of r< and r i} and r b is the greater. 
PJ^cos #) is an associated Legendre function, discussed in 
Section 196. Using this expansion we obtain for the <p part of 

the above integral f *f r e i{Pm i^i^ ) ^e i{Pm 'i^i^ ) ^d(p % d(p h in which 

mi is associated with u^(i) f m\ with u^j), Pmi with Pu{(i), and 
Pm\ with Put(j). This vanishes unless Pmi — mi + m = 
and Pmi — m\ — m = 0; i.e., unless Pmi + Pm\ = mi + m\. 

1 For a proof of this see J. H. Jeans, "Electricity and Magnetism," 5th ed„ 
Equations 152 and 196, Cambridge University Press, 1927. 



242 MANY-ELECTRON ATOMS [IX-30d 

This completes the proof of the theorem that H mn = unless 
Xmi is the same for \f/ n and ^ m . 

Of the non-vanishing elements H mn only certain of the diagonal 
ones need to be evaluated in order to calculate the energy levels, 
as we have seen in the last section. Because of the orthogonality 
of the u'a, Equation 30-13 vanishes unless P = 1 (the identity 
operation) when a diagonal element H mm is being considered. 
Since the u y s are also normalized, this expression reduces to 

%futti)f(ri)u t (i)d Ti = J/,-, (30-17) 

t X 

a relation which defines the quantities I,. 

Similarly, the orthogonality of the u's restricts P in Equation 
30-15 to P = 1 and P = (ij), the identity operation and the 
interchange of i and j, respectively. The first choice of P 
contributes the terms 

2 Jti r *(t>f0^W«f0')*4r / ~ 2« 7 <» (30-18) 
while the second yields 



- 2 J ^W^O")^ (t>r0>*n*/ = ~ 2*"' (3 °~ 19) 

»J>* *J>» 

The integral if <; vanishes unless the spins of w f (i) and i^O') are 
parallel, i.e., unless m.< = m 8] . 

The functions /* reduce to integrals over the radial part of 
u(i), 

h = JRtiiriWdRniirddu. (30-20) 

We shall not evaluate these further. 

The functions J», and Ka may" be evaluated by using the 
expansion for 1/nj given in Equation 30-16. For J a the (pi 

part of the integral has the form f T e im *id<p iy which vanishes 

unless m = 0. The double sum in the expansion 30-16 thus 
reduces to a single sum over k, which can be written 

J a = %a k (lmi; Vm[)F k (nl; n'V), (30-21) 

k 

in which nlmi and n'Vm\ are the quantum numbers previously 
represented by n< and n y , respectively. a k and F* are given by 



IX-30d] SLATER'S TREATMENT OF COMPLEX ATOMS 243 

aHlmrl'm') - (2 * ± 1)( * - |m ' l)! (2Z ' ± M Z ND' 
a (fcm, £ m,) - 2(z + |m ^ f 2 (J' + |mf|)! 

jT T {P^'(cos t ) ) 2 P*°(cos #»•) sin #«Z0,- 

jT x {P^'i(cos #,) } 2 P°(cos *,) sin tfyto/, (30-22) 

and 

F*(nl; nT) = (^) 2 e 2 Jj " J[ ^S/^O^^rCry)^?^^,, (30-23) 

The a's are obtained from the angular parts of the wave functions, 
which are the same as for the hydrogen atom (Tables 21-1 and 
21-2, Chap. V). Some of these are given in Table 30-3, taken 

Table 30-3. — Values of a k (lmi; I'm'i) 
(In cases with two ± signs, the two can be combined in any of the four 

possible ways) 



Electrons 


I 


rrt\ 


V 


mj 


a° 


a 2 


a* 


88 














1 








sp 








1 


±1 


















1 













VV 


1 


±1 


1 


±1 




K 5 







1 


±1 


1 







-%5 







1 





1 







^5 





8d 








2 


±2 


















2 


±1 


















2 













pd 




±1 


2 


±2 




%5 









±1 


2 


±1 




-Ms 









±1 


2 







-«5 












2 


±2 




"«6 












2 


±1 




^5 












2 







^5 





dd 


2 


±2 


2 


±2 




**9 


J*41 




2 


±2 


2 


±1 




-%> 


-«41 




2 


±2 


2 







-«9 


%41 




2 


±1 


2 


±1 




K9 


J %41 




2 


±1 


2 







K9 


- 2 **41 




2 





2 







Yt* 


8 $441 



244 MANY-ELECTRON ATOMS [IX-80e 

from Slater's paper. The P's, on the other hand, depend on the 
radial parts of the wave functions, which for the best approxima- 
tion are not hydrogenlike. 

Kij may be similarly expressed as 

Ku = Xb\lmi) I'mWinl; nT), (3(H2 4) 

k 

in which 

b k {lmi)Vm[) = 

(t - \mi - m{|)l(2i + l)(l - h^KgT + l)(i' - \m[\) \ 
4(t + |m, -mj|)!(l + |m«|)(r + |m{|)! 

j Ppj^cos ^P^ 1 (cos tf)PJpr-tf (cos #) sin &» | *, (30-25) 
and 
G k (nl;n'V) = e 2 (4ir)*J[ <D j[ <D 2J»,(n)B»^(r < )B»i(ry)12»*r(r / ) 

^rtfdr«dr,. (30-26) 

The functions 6* are given in Table 30-4. The functions G k 
are characteristic of the atom. 

30e. Empirical Evaluation of Integrals. Applications. — We 
have now carried the computations to a stage at which the 
energy levels may be expressed in terms of certain integrals 
Ii, F k ) and G k which involve the radial factors of the wave 
functions. One method of proceeding further would be to assume 
some form for the central field fl(r t ), determine the functions 
Rmi(ri), and use them to evaluate the integrals. However, 
another and simpler method is available for testing the validity 
of this approximation, consisting in the use of the empirically 
determined energy levels to evaluate the integrals, a check on 
the theory resulting from the fact that there are more known 
energy levels than integrals to be determined. 

For example, if we substitute for ff n, etc., for the configuration 
np 2 the expression in terms of J t , F k , and G k , using the results of 
the previous section and Equation 30-10, we obtain for the 
energies of the terms 1 Z), 3 P, and X S the quantities 

>D:TF = 2/(n, 1) + F« + % 5 F\ 

V: W = 2/(n, 1) + F° - Y 2b F* - % 6 Q*, 

*S: W = 2/(n, 1) + F° + W + %5<? 2 - 



IX-30e] SLATER'S TREATMENT OF COMPLEX ATOMS 



245 



Table 30-4. — Values of 6*(Zmj; j'mj) 
(In cases where there are two ± signs, the two upper, or the two lower, 
signs must be taken together) 



Electrons 


I 


mi 


i 


m'j 


b Q 


6 1 


6* 


6 s 


b* 


88 














1 














8p 








i 


±1 





X 



















i 








X 











VV 


1 


±1 


i 


±1 


1 





«5 










1 


±1 


i 











K 5 










1 


±1 


i 


+ 1 








^5 










1 





i 





1 





^5 








sd 








2 


±2 








X 
















2 


±1 








X 
















2 











X 








pd 




±1 


2 


±2 





X 





%45 









±1 


2 


±1 





h 





%45 









±1 


2 








X* 





x x« 









±1 


2 


+ 1 











3 %45 









±1 


2 


+ 2 











4 ^45 












2 


±2 











^45 












2 


±1 





X 





2 ^45 












2 








Vl5 





2 %45 





dd 


2 


±2 


2 


±2 


1 





H9 





Ha 




2 


±2 


2 


±1 








%9 





%41 




2 


±2 


2 











n* 





15 A*i 




2 


±2 


2 


Tl 














35 441 




2 


±2 


2 


=F2 














7 ^41 




2 


±1 


2 


±1 


1 





X9 





*9i41 




2 


±1 


2 











X9 





*%4l 




2 


±1 


2 


Tl 








%9 





4 %41 




2 





2 





1 





%9 





3 %41 



Examination of Equations 30-18 and 30-19 shows that for 
equivalent electrons F is equal to G (with the same index). We 
therefore have for the separations of the levels for np 2 

iD -*P =% 5 F*(nl;nl), 
ifl- iD = ^ 5 F 2 (nl;nl). 



The theory therefore indicates that, if the approximations which 



246 MANY-ELECTRON ATOMS [IX-31 

have been made are valid, the ratio of these intervals should be 
2:3, a result which is obtained without the evaluation of any 
radial integrals at all. In addition, since F 2 is necessarily 
positive, this theory gives the order of the terms, 3 P lying lowest, 
l D next, and l S highest. This result is in agreement with 
Hund's empirical rules, that terms with largest multiplicity 
usually lie lowest, and that, for a given multiplicity, terms 
with largest L values usually lie lowest. 1 

Slater gives the example of the configuration 2 ls 2 2s 2 2p 6 
3s 2 3p 2 of silicon, for which the observed term values 3 are 

3 P = 65615 cm" 1 , 
l D = 59466 cm" 1 , 
l S = 50370 cm" 1 , 

so that the ratio l D - 3 P to l S - l D is 2:2.96, in excellent 
agreement with the theory. In other applications, however, 
large deviations have been found, most of which have been 
explained by considering higher approximations based on the 
same general principles. 4 

31. VARIATION TREATMENTS FOR SIMPLE ATOMS 

The general discussion of Section 30, which is essentially a 
perturbation calculation, is not capable of very high accuracy, 
especially since it is not ordinarily practicable to utilize any 
central field except the coulombic one leading to hydrogenlike 
orbital functions. In this section we shall consider the applica- 
tion of the variation method (Sec. 26) to low-lying states 
of simple atoms such as lithium and beryllium. This type of 
treatment is much more limited than that of the previous 
section, but for the few states of simple atoms to which it has 
been applied it is more accurate. 

1 Pauling and Goudsmit, "The Structure of Line Spectra," p. 166. 

2 This configuration gives the same interval ratios as np 2 } only the absolute 
energy being changed by the presence of the closed shells. 

3 As mentioned in Section 5a, term values are usually given in cm -1 and 
are measured downward from the lowest state of the ionized atom. Hence 
the largest term value represents the lowest energy level. 

4 There have been many papers on this subject; a few are: C. W. Ufford, 
Phys. Rev. 44, 732 (1933); G. H. Shortley, Phys. Rev. 43, 451 (1933); M. H. 
Johnson, Jr., Phys. Rev. 43, 632 (1933); D. R. Inglis and N. Ginsburg, 
Phys. Rev. 43, 194 (1933). A thorough treatment is given by E. U. Condon 
and G. H. Shortlev, "The Theory of Atomic Spectra," Cambridge, 1935. 



IX-31a] VARIATION TREATMENTS FOR SIMPLE ATOMS 247 

The principles involved are exactly the same as those discussed 
in Section 26 and applied to helium in Section 29c, so we shall 
not discuss them further but instead study the different types of 
variation functions used and the results achieved. 

31a. The Lithium Atom and Three -electron Ions. — Table 31-1 
lists the variation functions which have been tried for the 
lowest state of lithium, which has the configuration ls 2 2s. All 
these functions are of the determinant type given in Equation 
30-7 and in all of them the orbital part of u u (i) is of the form 

e ao , in which Z', the effective atomic number for the K shell, 
is one of the parameters determined by the variation method. 
The table gives the expressions for 6, the orbital part of u 2a (i), 
the function for the 2s electron. In addition, the upper limit 
to the total energy of the atom is given, and also the value of the 
first ionization potential calculated by subtracting the value of 
the energy calculated for Li + from the total energy calculated 
for Li. The Li + calculation was made with the use of the same 
type of Is function used in Li for the K shell, in order to cancel 
part of the error introduced by this rather poor K function. 
The table also gives the differences between these calculated 
quantities and the experimental values. 



Table 31-1. — Variation Functions for the Normal Lithium Atom 

Units: RJw 
Experimental total energy: —14.9674; experimental ionization potential: 

0.3966 



2s function 1 


Total 
energy 


Differ- 
ence 


Ionization 
potential 


Differ- 
ence 


1. 


r 


-14.7844 


0.1830 


0.3392 


0.0574 


2. 


b = re ^ 


-14.8358 


.1316 


.3906 


.0060 


3. 


b = e ~%i- 1 ) 


-14.8366 


.1308 


.3912 


.0054 


4. 


r - rr— - f — 

b - a-e O0 - e ao 

ao 


-14.8384 


.1290 


.3930 


.0036 



i The function 1 was used by C. Eckart, Pkya. Rev. 36, 878 (1930), 2 and 3 by V. G ille- 
min and C. Zener, Z. /. Phya. 61, 199 (1930), and 4 by E. B. Wilson, Jr., /. Chem. Phya. 1, 
211 (1933). The last paper includes similar tables for the ions Be + , B ++ , and C +++ . 



248 



MANY-ELECTRON ATOMS 



[IX-31a 



Table 31-2 lists the best values of the parameters for these 
lithium variation functions. Figure 31-1 shows the total 
electron distribution function 4xr 2 p for lithium, calculated using 

Table 31-2. — Parameter Values for Lithium Variation Functions 

Z'r 







Mli 


= e ao 








Function 


Z' 


V 


. 


r 


1 


b = e Hv 1 ) 




2 686 


0.888 








\ do / 






2. 


r 

b = re ^ 




2.688 


.630 






3. 


b = e °°( a. 1 ) 




2 . 688 


.630 


5 56 




4. 


b = or-e <* - e ao . .. 




2.69 


.665 


1.34 


1.5 



the best of these functions, p is the electron density, which can 
be calculated from f in the following manner: 



P = 3JW*-idr*. 



(31-1) 



\p*\f/dTidT 2 dn gives the probability of finding electron 1 in the 
volume element dri, electron 2 in dr 2 , and electron 3 in dr z . 
Integration over the coordinates of electrons 1 and 2 gives the 
probability of finding electron 3 in dr 3 , regardless of the positions 
of 1 and 2. Since \f/*\p is symmetric in the three electrons, the 
probability of finding one electron in a volume element dxdydz in 
ordinary three-dimensional space is three times the probability of 
finding a particular one. Figure 31-1 shows clearly the two shells 
of electrons in lithium, the well-marked K shell and the more 
diffuse L shell. Due to the equivalence of the three electrons, 
we cannot say that a certain two occupy the K shell and the 
remaining one the L shell, but we can say that on the average 
there are two electrons in the K shell and one in the L shell. 

The next step to be taken is to apply a variation function to 
lithium which recognizes explicitly the instantaneous, instead 
of just the average, influence of the electrons on each other. 
Such functions were found necessary to secure really accurate 
results for helium (Sec. 29c), but their application to lithium 



IX-81b] VARIATION TREATMENTS FOR SIMPLE ATOMS 249 

involves extremely great complications. This work has been 
begun (by James and Coolidge at Harvard 1 ). 

31b. Variation Treatments of Other Atoms. — Few efforts have 
been made to treat more complicated atoms by this method. 
Beryllium has been studied by several investigators but the 
functions which give good results for lithium are not nearly so 
accurate for heavier atoms. Hydrogenlike functions with 
variable effective nuclear charges (function 1 of Table 31-1 is 




Fig. 31-1. — The electron distribution function D = 4xr 2 p for the normal lithium 

atom. 

such a function for n = 2, I = 0) have been applied to the case 
of the carbon atom, 2 the results being in approximate agreement 
with experiment. Functions of the types 2 and 3 of Table 31-1 
have also been tried 3 for Be, B, C, N, 0, F, and Ne. A more 
satisfactory attack has been begun by Morse and Young, 4 who 
have prepared numerical tables of integrals for wave functions 
dependent on four parameters (one for Is, two for 2s, and one 

1 Private communication to the authors; see H. M. James and A. S. 
Coolidge, Phys. Rev. 47, 700 (1935), for a preliminary report. 

1 N. F. Bbakdslby, Phys. Rev. 39, 913 (1932). 

8 C. Zbner, Phys. Rev. 36, 51 (1930). 

4 P. M. Morse and L. A. Young, unpublished calculations (available av 
the Massachusetts Institute of Technology). 



250 MANY-ELECTRON ATOMS [IX-32a 

for 2p one-electron functions) for the treatment of the K and L 
shells of atoms. 

The analytical treatment of complicated atoms by this method 
is at present too laborious for the accuracy obtained, but it may 
be possible to find new forms for the variation function which will 
enable further progress to be made. 

32. THE METHOD OF THE SELF-CONSISTENT FIELD 

The previous sections give some indication of the difficulty 
of treating many-electron atoms in even an approximate manner. 
In this section we shall discuss what is probably the most success- 
ful effort which has yet been made in attacking this problem, 
at least for those atoms which are too complicated to treat by 
any satisfactory variation function. Both the principle and 
the difficult technique involved are due to Hartree, 1 who, with the 
aid of his students, has now made the numerical computations 
for a number of atoms. In Section 326 we shall show the 
connection between this method and those previously discussed. 

32a. Principle of the Method. — In Section 306 we have 
pointed out that the wave equation for a many-electron atom 
can be separated into single-electron wave functions not only 
when the mutual interactions of the electrons are completely 
neglected but also when a central field v(x t ) for each electron is 
added to the unperturbed equation and subtracted from the 
perturbation term. Each of the resulting separated unper- 
turbed wave equations describes the motion of an electron in a 
central field which is independent of the coordinates of the 
other electrons. The perturbation treatment considered in 
Section 30 was based on the idea that a suitable choice could be 
made of these central fields for the individual electrons so that 
they would represent as closely as possible the average effect 
upon one electron of all the other electrons in the atom. 

The important step in the application of such a method of 
treatment is the choice of the potential-energy functions repre- 
senting the central fields. The assumption made by Hartree 
is that the potential-energy function for one electron due to a 
second electron is determined approximately by the wave 
function for the second electron, m^(2), say, being given by the 

1 D. R. Hartree, Proc. Cambridge Phil Soc. 24, 89, 111, 426 (1928). 



IX-32a] THE METHOD OF THE SELF-CONSISTENT FIELD 251 

potential corresponding to the distribution of electricity deter- 
mined by the probability distribution function u$(2) u fi (2). 
This is equivalent to assuming that the wave function for the 
second electron is independent of the coordinates of the first 
electron. The complete central-field potential-energy function 
for the first electron is then obtained by adding to the potential- 
energy function due to the nucleus those potential-energy 
functions due to all the other electrons, calculated in the way 
just described. The wave function for the first electron can 
then be found by solving the wave equation containing this 
complete potential-energy function. 

It is seen, however, that in formulating a method of calculating 
the functions u^(k) for an atom we have assumed them to be 
known. In practice there is adopted a method of successive 
approximations, each cycle of which involves the following 
steps : 

1. A potential-energy function due to the nucleus and all of 
the electrons is estimated. 

2. From this there is subtracted the estimated contribution 
of the fcth electron, leaving the effective potential-energy function 
for this electron. 

3. The resulting wave equation for the fcth electron is then 
solved, to give the wave function u^(k). Steps 2 and 3 are 
carried out for all of the electrons in the atom. 

4. Using the functions u^(k) obtained by step 3, the potential- 
energy functions due to the various electrons are calculated, 
and compared with those initially assumed in steps 1 and 2. 

In general the final potential-energy functions are not identical 
with those chosen initially. The cycle is then repeated, using 
the results of step 4 as an aid in the estimation of new potential- 
energy functions. Ultimately a cycle may be carried through 
for which the final potential-energy functions are identical 
(to within the desired accuracy) with the initial ones. The 
field corresponding to this cycle is called a self-consistent field 
for the atom. 

It may be mentioned that the potential-energy function due 
to an $ electron is spherically symmetrical, inasmuch as the 
probability distribution function u^u na is independent of <p and #. 
Moreover, as a result of the theorem of Equation 21-16 the 
potential-energy function due to a completed shell of electrons 



252 MANY-ELECTRON ATOMS [IX-82b 

is also spherically symmetrical. Spherical symmetry of the 
potential function greatly increases the ease of solution of the 
wave equation. 

Hartree employs the method of numerical integration sketched 
in Section 27c to solve the single-electron wave equations. 
In addition he makes the approximation of considering all 
contributions to the field as spherically symmetrical. Thus 
if some electron (such as a p electron) gives rise to a charge 
distribution which is not spherically symmetrical, this is averaged 
over all directions. Finally, the simple product of Equation 
30-2 is used for the wave function for the whole atom. As we 
have seen, this does not have the correct symmetry required by 
Pauli's principle. The error due to this involves the interchange 
energies of the electrons (Sec. 32c). 

32b. Relation of the Self-consistent Field Method to the 
Variation Principle. — If we choose a variation function of the 
form 

<t> = t*«(l) u p (2) • • • u y (N) (32-1) 

and determine the functions u^(i) by varying them individually 
until the variational integral in Equation 26-1 is a minimum, 
then, as shown in Section 26a, these are the best forms for the 
functions %(t) to use in a wave function of this product type for 
the lowest state. Neglecting the fact that Hartree averages 
all fields to make them spherically symmetrical, we shall now 
show 1 that the variation-principle criterion is identical, for this 
type of <£, with the criterion of the self-consistent field. If 
we keep each u^(i) normalized, then f<t>*<t>d,T = 1 and 

E = f<t>*H<txlT. (32-2) 

The operator H may be written as • 

H = 2 H < + 2$ (32 - 3) 

with 



*,;>t 



h 2 Ze 2 

Hi = -«4-v? - — ' (32-4) 

87T 2 m Ti v 

1 J. C. Slater, Phys. Rev. 35, 210 (1930); V. Fock, Z. f. Phys. 61, 126 
(1930). 



EC-82b] THE METHOD OF THE SELF-CONSISTENT FIELD 253 
Using this and the expression for <t> in Equation 32-1, we obtain 

\?(i)ut(j)^u { (i)u e (j)dT4Ti. (32-5) 



2J/' 



The variation principle can now be applied. This states that 
the best form for any function u^(i) is the one which makes E 
a minimum (keeping the function normalized). For this mini- 
mum, a small change 8u^(i) in the form of U[(i) will produce no 
change in E\ that is 8E = 0. 

The relation between 8u^(i) and 8E is 



-.JV 



8E = 8\ u?(i)H&t(i)dTi + 



sw 



■?(*)*?(J)tr*r(i)«iU)dT4T t , (32-6) 



in which the prime on the summation sign indicates that the 
term with j = i is not included. Let us now introduce the new 
symbol F if defined by the equation 

F i = H i + ^ f t*f (jJ^WfO')*-/, (32-7) 

or 

Fi = H { + V i9 
in which 

y 

Fi is an effective Hamiltonian function for the ith electron, and 
Vi the effective potential-energy function for the ith electron 
due to its interaction with the other electrons in the atom. 
Using the symbol Fi, we obtain as the condition that E be sta- 
tionary with respect to variation in u^(i) the expression (Eq. 
32-6) 

8E = 6fu?(i)F<UtWn - 0. (32-9) 



254 MANY-ELECTRON ATOMS [IX-32c 

A similar condition holds for each of the N one-electron functions 
tta(l), • • • , u,(N). 

Let us now examine the criterion used in the method of the 
self-consistent field. In this treatment the wave function 
U[(i) is obtained as the solution of the wave equation 

VfwrM + ^r(*i + ™ - ^)%« = 0, (32-10) 

or, introducing the symbol F if 

FMi) = e x Ui(i). (32-11) 

We know, however, that a normalized function u^(i) satisfying 
this equation also satisfies the corresponding variational equation 

h$u?(i)F x u ; {i)dT x = 0. (32-12) 

Equations 32-9 and 32-12 are identical, so that by using the 
variation method with a product-type variation function we 
obtain the same single-electron functions as by applying the 
criterion of the self-consistent field. 

32c. Results of the Self-consistent Field Method. — Hartree 
and others have applied the method of the self-consistent field 
to a number of atoms and ions. In one series of papers 1 Hartree 
has published tables of values of single-electron wave functions 
for Cl~, Cu + , K + , and Rb + . These wave functions, as given, 
are not normalized or mutually orthogonal, but values of the 
normalizing factors are reported. For these atoms the total 
energy has not been calculated, although values of the individual 
€t's are tabulated. (The sum of these is not equal to the total 
energy, even if interchange is neglected.) For O, + , ++ , and 
+++ , Hartree and Black 2 have given not only the wave functions 
but also the total energies calculated by inserting these single- 
electron wave functions into a determinant such as Equation 
30-7 and evaluating the integral E '= f\p*H\pdT. 

Several other applications 3 have been made of this method and 
a considerable number are now in progress. Slater 4 has taken 
Hartree's results for certain atoms and has found analytic expres- 

1 D. R. Hartree, Proc. Roy. Soc. A 141, 282 (1933); A 143, 506 (1933). 

2 D. R. Hartree and M. M. Black, Proc. Roy. Soc. A 139, 311 (1933). 

3 F. W. Brown, Phys. Rev. 44, 214 (1933); F. W. Brown, J. H. Bartlett, 
Jr., and C. G. Dunn, Phys. Rev. 44, 290 (1933); J. McDougall, Proc. Roy. 
Soc. A 138, 550 (1932); C. C. Torrance, Phys. Rev. 46, 388 (1934). 

* J. C. Slater, Phys. Rev. 42, 33 (1932). 



IX-32c] THE METHOD OF THE SELF-CONSISTENT FIELD 255 

sions for the single-electron wave functions which fit these results 
fairly accurately. Such functions are of course easier to use than 
numerical data. 

The most serious drawback to Hartree's method is probably 
the neglect of interchange effects, i.e., the use of a simple product- 




Fxo. 32-1. — The electron distribution function D for the normal rubidium 
atom, as calculated: I, by Hartree's method of the self-consistent field; II, by the 
screening-constant method; and III, by the Thomas-Fermi statistical method. 

type wave function instead of a properly antisymmetric one. 
This error is partially eliminated by the procedure of Hartree 
and Black described above, but, although in that way the energy 
corresponding to a given set of functions %(fc) is properly calcu- 
lated, the functions %(&) themselves are not the best obtainable 
because of the lack of antisymmetry of $. Fock 1 has considered 
this question and has given equations which may be numerically 
solved by methods similar to Hartree's, but which include inter- 
change. So far no applications have been made of these, but 
several computations are in progress. 2 

Figures 32-1, from Hartree, shows the electron distribution 
function for Rb+ calculated by this method, together with those 
given by other methods for comparison. 

1 V. Fock, Z. f. Phys. 61, 126 (1930). 

'See D. R. Hartree and W. Hartree, Proc. Roy. Soc. A 150, 9 (1936). 



256 MANY-ELECTRON ATOMS [IX-33a 

Problem 32-1. (a) Obtain an expression for the potential due to an 
electron in a hydrogenlike Is orbital with effective atomic number Z' - 
21 A&' (&) Using this result, set up the wave equationfor one electron in a 
helium atom in the field due to the nucleus and the other electron (assumed 
to be represented by the wave function mentioned above). Solve the wave 
equation by the method of difference equations (Sec. 27 d), and compare 
the resultant wave function with that chosen initially. 



33. OTHER METHODS FOR MANY-ELECTRON ATOMS 

Besides the methods discussed in the previous sections there 
are others yielding useful results, some of which will be briefly 
outlined in the following sections. Several methods have been 
proposed which are beyond the scope of this book, notably the 
Dirac^Van Vleck 2 vector model, which yields results similar to 
those given by the method of Slater of Section 30. 

33a. Semi-empirical Sets of Screening Constants. — One of the 
methods mentioned in Section 316 consists in building up an 
approximate wave function for an atom by the use of hydrogen- 
like single-electron functions with effective nuclear charges 
determined by the variation method. Instead of giving the 
effective atomic number Z', it is convenient to use the difference 
between the true atomic number and the effective atomic num- 
ber, this difference being called the screening constant. Pauling 3 
has obtained sets of screening constants for all atoms, not by 
the application of the variation method (which is too laborious), 
but by several types of reasoning based in part on empirical 
considerations, involving such quantities as x-ray term values 
and molecular refraction values. It is not< to be expected 
that wave functions formed in this manner will be of very great 
accuracy, but for many purposes they are sufficient and for many 
atoms they are the best available! The results obtained for 
Rb + are shown in Figure 32-1. 

Slater 4 has constructed a similar table, based, however, on 
Zener's variation-method calculations for the first ten elements 
(Sec. 316). His screening constants are meant to be used in 

1 P. A. M. Dirac, "The Principles of Quantum Mechanics," Chap. XI. 
* J. H. Van Vleck, Phys. Rev. 46, 405 (1934). 

8 L. Pauling, Proc. Roy. Soc. A 114, 181 (1927); L. Paulino and J. 
Sherman, Z. f. Krist. 81, 1 (1932). 

4 J. C. Slater, Phys. Rev. 36, 57 (1930). 



IX-33b] OTHER METHODS FOR MANY-ELECTRON ATOMS 257 

functions of the type r n -e~ z ' r instead of in hydrogenlike func- 
tions, the exponent n' being an effective quantum number. 

A discussion of an approximate expression for the wave func- 
tion in the outer regions of atoms and ions and its use in the 
treatment of various physical properties (polarizability, ioniza- 
tion potentials, ionic radii, etc.) has been given by Wasastjerna. 1 

33b. The Thomas -Fermi Statistical Atom. — In treating a 
system containing a large number of particles statistical methods 
are frequently applicable, so that it is natural to see if such 
methods will give approximate results when applied to the 
collection of electrons which surround the nucleus of a heavy 
atom. Thomas 2 and Fermi 3 have published such a treatment. 
In applying statistical mechanics to an electron cloud, it was 
recognized that it is necessary to use the Fermi-Dirac quantum 
statistics, based on the Pauli exclusion principle, rather than 
classical statistics, which is not even approximately correct for 
an electron gas The distinctions between these have been men- 
tioned in Section 296 and will be further discussed in Section 49. 

The statistical treatment of atoms yields electron distributions 
that are surprisingly good in view of the small number of electrons 
involved. These results have been widely used for calculating 
the scattering power of an atom for z-rays and for obtaining an 
initial field for carrying out the self-consistent-field computations 
described in the previous section. However, the Thomas-Fermi 
electron distribution does not show the finer features, such as 
the concentration of the electrons into shells, which are character- 
istic of the more refined treatments. Figure 32-1 shows how the 
Thomas-Fermi results compare with Hartree's and Pauling's 
calculations for Rb+ . 

General References on Line Spectra 

Introductory treatments : 

L. Pauling and S. Goudsmit: "The Structure of Line Spectra," McGraw- 
Hill Book Company, Inc., New York, 1930. 

H. E. White: "Introduction to Atomic Spectra," McGraw-Hill Book 
Company, Inc., New York, 1934. 

1 J. A. Wasastjerna, Soc. Scieni. Fennica Comm. Phys.-Math., vol. 6, 
Numbers 18-22 (1932). 

*L. H. Thomas, Proc. Cambridge Phil. Soc. 23, 642 (1927). 
3 E. Fermi, Z. /. Phys. 48, 73; 49, 560 (1928). 



258 MANY-ELECTRON ATOMS [IX-33b 

A. E. Ruark and H. C. Urey: "Atoms, Molecules and Quanta," 
McGraw-Hill Book Company, Inc., New York, 1930. 

A thorough quantum-mechanical treatment: 

E. U. Condon and G. H. Shortley: "The Theory of Atomic Spectra," 
Cambridge University Press, 1935. 

Tabulation of term values: 

R. F. Bacher and S. Goudsmit: "Atomic Energy States," McGraw- 
Hill Book Company, Inc., New York, 1932. 



CHAPTER X 
THE ROTATION AND VIBRATION OF MOLECULES 

The solution of the wave equation for any but the simplest 
molecules (some of which are discussed in Chap. XII) is a very 
difficult problem. However, the empirical results of molecular 
spectroscopy show that in many cases the energy values bear a 
simple relation to one another, such that the energy of the 
molecule (aside from translational energy) can be conveniently 
considered to be made up of several parts, called the electronic 
energy, the vibrational energy, and the rotational energy. 
This is indicated in Figure 34-1, showing some of the energy 
levels for a molecule of carbon monoxide, as calculated from the 
observed spectral lines by the Bohr frequency rule (Sec. 5a). 
It is seen that the energy levels fall into widely separated 
groups, which are said to correspond to different electronic states 
of the molecule. For a given electronic state the levels are 
again divided into groups, which follow one another at nearly 
equal intervals. These are said to correspond to successive 
states of vibration of the nuclei. Superimposed on this is the 
fine structure due to the different states of rotation of the mole- 
cule, the successive rotational energy levels being separated by 
larger and larger intervals with increasing rotational energy. 
This simplicity of structure of the energy levels suggests that it 
should be possible to devise a method of approximate solution 
of the wave equation involving its separation into three equa- 
tions, one dealing with the motion of the electrons, one with 
the vibrational motion of the nuclei, and one with the rotational 
motion of the nuclei. A method of this character has been 
developed and is discussed in the following section. The 
remaining sections of this chapter are devoted to the detailed 
treatment of the vibrational and rotational motion of molecules 
of various types. 

34. THE SEPARATION OF ELECTRONIC AND NUCLEAR MOTION 

By making use of the fact that the mass of every atomic nucleus 
is several thousand times as great as the mass of an electron, 

259 



260 THE ROTATION AND VIBRATION OF MOLECULES [X-34 

Born and Oppenheimer 1 were able to show that an approximate 
solution of the complete wave equation for a molecule can be 
obtained by first solving the wave equation for the electrons 



cm"' 










90,000 
















80,000 
70,000 


- 






60,000 








50,000 








40,000 
30,000 


- 


Etc.* 
















E+c.+ 












r 


20,000 

















s~< 




































10,000 


— 






























v> 




















Fig. 34r-l. — Energy levels for the carbon monoxide molecule. On the left are 
shown various electronic levels, with vibrational fine structure for the normal 
state, and on the right, with one hundred fold increase of scale, the rotational 
fine structure for the lowest vibrational level. 

alone, with the nuclei in a fixed configuration, and then solving 
a wave equation for the nuclei alone, in which a characteristic 
energy value of the electronic wave equation, regarded as a 
1 M. Bokn and J. R. Oppbnhbimbb, Ann. d. Phya. 84, 457 (1927). 



X-34] ELECTRONIC AND NUCLEAR MOTION 261 

function of the internuclear distances, occurs as a potential 
function. Even in its simplest form the argument of Born and 
Oppenheimer is very long and complicated. On the other 
hand, the results of their treatment can be very simply and 
briefly described. Because of these facts, we shall content 
ourselves with describing their conclusions in detail. 

The complete wave equation for a molecule consisting of r 
nuclei and s electrons is 

r * 

y-i *-i 

in which M } is the mass of the jth nucleus, m the mass of each 
electron, v ; ? the Laplace operator in terms of the coordinates 
of the jth nucleus, and v» ? the same operator for the ith electron. 
V is the potential energy of the system, of the form 






the sums including each pair of particles once. Here Z, is the 
atomic number of the jth nucleus. 

Let us use the letter £ to represent the 3r coordinates of the 
r nuclei, relative to axes fixed in space, and the letter x to repre- 
sent the 3s coordinates of the s electrons, relative to axes deter- 
mined by the coordinates of the nuclei (for example, as described 
in Section 48). Let us also use the letter v to represent the 
quantum numbers associated with the motion of the nuclei, 
and n to represent those associated with the motion of the 
electrons. The principal result of Born and Oppenheimer^ 
treatment is that an approximate solution ^«,„(x, £) of Equation 
34-1 can be obtained of the form 

4>nA*> f) = *n(*> £)*».,(£). (34r-2) 

The different functions \l/ n (x, £), which may be called the 
electronic wave functions, correspond to different sets of values 
of the electronic quantum numbers n only, being independent 
of the nuclear quantum numbers v. On the other hand, each of 
these functions is a function of the nuclear coordinates J as 
well as the electronic coordinates x. These functions are 



262 THE ROTATION AND VIBRATION OF MOLECULES [X-84 

obtained by solving a wave equation for the electrons alone, the 
nuclei being restricted to a fixed configuration. This wave equation 
is 



8w 2 m { 



2 *#«(*, «) + ^°{ V.(& - V{z, {)}*„<*, |) = 0. (34-3) 



U(r)t 



1 



It is obtained from the complete wave equation 34-1 by omitting 
the terms involving v/, replacing ^ by ^ n (x f £), and writing 
U»(£) in place of W. The potential function V(x, £) is the 

complete potential function of 
Equation 34-1. It is seen that 
for any fixed set of values of 
the s nuclear coordinates £ this 
equation 3^-3, which we may 
call the electronic wave equation, 
is an ordinary wave equation for 
the s electrons, the potential- 
energy function V being depend- 
ent on the values selected for the 
nuclear coordinates £. In con- 
sequence the characteristic elec- 
Fig. 34-2.— A typical function u(r) for tronic energy values U n and the 

a diatomic molecule (Morae function). , , - , . . 

electronic wave functions \j/ n 
will also be dependent on the values selected for the nuclear 
coordinates; we accordingly write them as (/»($) and yp n {x } £). 
The first step in the treatment of a molecule is to solve this 
electronic wave equation for all configurations of the nuclei. 
It is found that the characteristic values U n (£) of the electronic 
energy are continuous functions of the nuclear coordinates £. 
For example, for a free diatomic molecule the electronic energy 
function for the most stable electronic state (n = 0) is a function 
only of the distance r between the two nuclei, and it is a con- 
tinuous function of r, such as shown in Figure 34-2. 

Having evaluated the characteristic electronic energy £/ n (£) 
as a function of the nuclear coordinates £ for a given set of 
values of the electronic quantum numbers n by solving the 
wave equation 34-3 for various nuclear configurations, we next 
obtain expressions for the nuclear wave functions ^ n ,*({). It 
was shown by Born and Oppenheimer that these functions are 



X-35] VIBRATION OF DIATOMIC MOLECULES 263 

the acceptable solutions of a wave equation in the nuclear 
coordinates £ in which the characteristic electronic energy 
function U n (£) plays the role of the potential energy; that is, 
the nuclear wave equation is 

r 

2i V ' ¥n '" a) + 1? {W "" ~ u »W+».>ti) = 0. (34-1) 
y-i 

There is one such equation for each set of values of the electronic 
quantum numbers n, and each of these equations possesses an 
extensive set of solutions, corresponding to the allowed values 
of the nuclear quantum numbers v. The values of W n , v are the 
characteristic energy values for the entire molecule; they depend 
on the electronic and nuclear quantum numbers n and v. 

The foregoing treatment can be formally justified by a pro- 
cedure involving the expansion of the wave functions and 
other quantities entering in the complete wave equation 34-1 
as power series in (m /M) H , in which M is an average nuclear 
mass. The physical argument supporting the treatment is 
that on account of the disparity of masses of electrons and nuclei 
the electrons carry out many cycles of their motion in the time 
required for the nuclear configuration to change appreciably, 
and that in consequence we are allowed to quantize their motion 
for fixed configurations (by solving the electronic wave equation), 
and then to use the electronic energy functions as potential energy 
functions determining the motion of the nuclei. 

When great accuracy is desired, and in certain cases when 
only ordinary accuracy is required, it is necessary to consider the 
coupling between electronic and nuclear motions, and especially 
between the electronic angular momentum (either spin or 
orbital) and the rotation of the molecule. We shall not discuss 
these questions, 1 but shall treat only the simplest problems in 
the complex field of molecular structure and molecular spectra 
in the following sections. Some further discussion is also 
given in Chapter XII and in Section 48 of Chapter XIV. 

35. THE ROTATION AND VIBRATION OF DIATOMIC MOLECULES 

In the previous section we have stated that an approximate 
wave function for a molecule can be written as a product of two 
1 See the references at the end of the chapter. 



264 THE ROTATION AND VIBRATION OF MOLECULES [X-36a 

factors, one a function of the electronic coordinates relative to 
the nuclei and the other a function of the nuclear coordinates. 
In this section we shall consider the nuclear function and the 
corresponding energy levels for the simplest case, the diatomic 
molecule, assuming the electronic energy function U n (r) to be 
known. 

35a. The Separation of Variables and Solution of the Angular 
Equations. — The wave equation for the rotation and vibration 
of a diatomic molecule (Eq. 34-4) has the form 



1 2 i i 1 9 i i °7T 



V?*n,„ + irVty** + ^{Wn,, - U n (r))* n , v = 0, (35-1) 



in which ^ n ,v= yfrn, v {x\, y\ y Zi, x 2y 2/2, £2) is the wave function for 
the nuclear motion, M\ and M 2 are the masses of the two nuclei, 
and 

x ly y xy and z x being the Cartesian coordinates of the tth nucleus 
relative to axes fixed in space. Equation 35-1 is identical with 
the wave equation for the hydrogen atom, the two particles 
here being the two nuclei instead of an electron and a proton. 
We may therefore refer to the treatment which has already 
been given of this equation in connection with hydrogen. All 
the steps are the same until the form for U n (r) is inserted into 
the radial equation. 

In Section 18a we have seen that Equation 35-1, expressed in 
terms of the Cartesian coordinates of the two particles, can be 
separated into two equations, one describing the translational 
motion of the molecule and the other its internal motion. The 
latter has the form 



+ 



r 2 dr\ drj^r 2 sin <3^ Sm d$) ^ r 2 sin 2 d<p 2 

^r\W- tf(r)}*«0, (35-3) 

in which /x, the reduced mass, is given by the equation 

M,M 2 



anc 



X-36a] VIBRATION OF DIATOMIC MOLECULES 265 

and r, #, <p are polar coordinates of the second nucleus relative to 
the first as origin. In Section 18a it was also shown that this 
equation can be separated into three equations in the three 
variables <p } #, and r, respectively. The solution of the <p and # 
equations, which are obtained in Sections 186, 18c, and 19, are 

*m(<p) = — ^ iA " (35-5) 

V27T 

in which P^cos #) is an associated Legendre function (Sec. 196), 
<£> and are the <p and # factors, respectively, in the product 
function 

iKr, #, <p) = #(r)e(t>)*(*). (35-7) 

Instead of the azimuthal quantum number I, used for the hydro- 
gen atom, we have here adopted the letter K, and for the magnetic 
quantum number m we here use M, in agreement with the usual 
notation for molecular spectra. Both M and K must be integers, 
for the reasons discussed in Sections 186 and 18c, and, as there 
shown, their allowed values are 

K = 0, 1, 2, • • • ; M = -K, -K + 1, • • , K - 1, K. 

(35-8) 

Just as in the case of hydrogen, the quantum numbers M and K 
represent angular momenta (see also Sec. 52), the square of the 
total angular momentum due to the rotation of the molecule 1 
being 

K(K + 1)£,, (35-9) 

while the component of this angular momentum in any specially 
chosen direction (taken as the z direction) is 

M~ (35-10) 

In Section 40d it will be shown that dipole radiation is emitted 
or absorbed only for transitions in which the quantum number 

1 There may be additional angular momentum due to the electrons. 



266 THE ROTATION AND VIBRATION OF MOLECULES [X-36b 

K changes by one unit; i.e., the selection rule for K is 

AK = ±1. 

Likewise, the selection rule for M is 

AM = or ±1. 

The energy of the molecule does not depend on M (unless there 
is a magnetic field present), so that this rule is not ordinarily 
of importance in the interpretation of molecular spectra. 
The equation for R(r) (Eq. 18-26) is 



^•t>+[- KJ ^ ±)+ ->- u ™ 



R = 0, 

(35-11) 



in which for simplicity wc have omitted the subscripts n and v. 
This may be simplified by the substitution 

R(r) = is(r), (35-12) 

which leads to the equation 



d*S 
dr* 



+ [- I ^±^+ S ^iW-U(r)}]s = 0. (35-13) 



35b. The Nature of the Electronic Energy Function. — The 

solution of the radial equation 35-13 involves a knowledge of 
the electronic energy function U(r) discussed in Section 34. The 
theoretical calculation of U(r) requires the solution of the wave 
equation for the motion of the electrons, a formidable problem 
which has been satisfactorily treated only for the very simplest 
molecules, such as the hydrogen molecule (Sec. 43). It is 
therefore customary to determine U(r) empirically by assuming 
some reasonable form for it involving adjustable parameters 
which are determined by a comparison of the observed and calcu- 
lated energy levels. 

From the calculations on such simple molecules as the hydrogen 
molecule and from the experimental results, we know that 
U(r) for a stable diatomic molecule is similar to the function 
plotted in Figure 34-2. When the atoms are very far apart 
(r large), the energy is just the sum of the energies of the two 
individual atoms. As the atoms approach one another there 



X-35c] VIBRATION OF DIATOMIC MOLECULES 267 

is for stable states a slight attraction which increases with 
decreasing r, as is shown by the curvature of U in Figure 34-2. 
For stable molecules, U must have a minimum value at the 
equilibrium separation r = r e . For smaller values of r, U rises 
rapidly, corresponding to the high repulsion of atoms "in contact/ 7 

For most molecules in their lower states of vibration it will be 
found that the wave function has an appreciable value only in a 
rather narrow region near the equilibrium position, this having 
the significance that the amplitude of vibration of most molecules 
is small compared to the equilibrium separation. This is impor- 
tant because it means that for these lower levels the nature of 
the potential function near the minimum is more important than 
its behavior in other regions. 

However, for higher vibrational levels, that is, for larger ampli- 
tudes of vibration, the complete potential function is of impor- 
tance. The behavior of U in approaching a constant value for 
larger values of r is of particular significance for these higher 
levels and is responsible for the fact that if sufficient energy is 
transferred to the molecule it will dissociate into two atoms. 

In the following sections two approximations for U(r) will be 
introduced, the first of which is very simple and the second 
somewhat more complicated but also more accurate. 

35c. A Simple Potential Function for Diatomic Molecules. — 
The simplest assumption which can be made concerning the force 
between the atoms of a diatomic molecule is that it is proportional 
to the displacement of the internuclear distance from its equilib- 
rium value r e . This corresponds to the potential function 

U(r) = V 2 k(r - r e )\ (35-14) 

which is plotted in Figure 35-1. k is the force constant for the 
molecule, the value of which can be determined empirically from 
the observed energy levels. A potential-energy function of this 
type is called a Hooke's-law potential energy function. 

It is obvious from a comparison of Figures 34r-2 and 35-1 
that this simple function is not at all correct for large internuclear 
distances. Nevertheless, by a proper choice of k a fair approxi- 
mation to the true U{r) can be achieved in the neighborhood 
of r = r e . This approximation corresponds to expanding the 
true U(r) in a Taylor series in powers of (r — r e ) and neglecting 
all powers above the second, a procedure which is justified only 



268 THE ROTATION AND VIBRATION OF MOLECULES [X-35c 



for small values of r — r . The coefficient of (r — r e )° (that is, 
the constant term) in this expansion can be conveniently set 
equal to zero without loss of generality so far as the solution 
of the wave equation is concerned. The linear term in the 



u(r) 




Fig. 36-1. — Hooke's-law potential function as an approximation to U (r). 

expansion vanishes, inasmuch as U(r) has a minimum at r = r 6 , 

\(d 2 Jj\ 
and so the series begins with the term o\~J~2j ( r ~" r «) 2 - 

Comparison with Equation 35-14 shows that the force constant 

k is equal to ( -j-^ ) 

Insertion of this form for U(r) into the radial equation 35-13 
yields the equation 



d*S 
dr* 



+ 



[- 



K( K + 1) , 8tt 

~2 I 



^ 



rfc(r - r e ) 



■}> - °> 



(35-15) 



which may be transformed by the introduction of the new inde- 
pendent variable p = r — r e (the displacement from the equilib- 
rium separation) into the equation 

** + *^iw - **„« - -»- K(K ± l) \s = 

Since the approximation which we have used for U(r) is good only 
for p small compared to r«, it is legitimate to introduce the 
expansion 



X-88c] VIBRATION OF DIATOMIC MOLECULES 269 

a step which leads to the result 

P + j^{ w - K ( K + IV + 2K ( K + Djtp - 

3K(K + \)~y - -kp^S = 0, (35-16) 

in which powers of p/r, greater than the second have been 
neglected, and the symbol a has been introduced, with 

and I e — v>r*. I e is called the equilibrium moment of inertia of 
the molecule. 

By making a suitable transformation p — $ -\- a, we can 
eliminate the term containing the first power in the independent 
variable, obtaining thereby an equation of the same form as 
Equation 11-1, the wave equation for the harmonic oscillator, 
which we have previously solved. It is easily verified that the 
proper value for a is 

SK(K + 1> + V 2 krf 

and that the introduction of this transformation into Equation 
35-16 yields the equation 



d*S S^(\ {K(K + !)„]* 1 

- [jfc + 3K(K + l)^]f 2 }-S = 0. (35-18) 

We seek the solutions of this equation which make ^(r, #, <p) of 
Equation 35-7 a satisfactory wave function. This requires 
that S vanish for r = and r = oo , the former condition entering 

because of the relation R = -S. We know the solutions of the 

r 

equation which vanish f or f = — oo and f = + <*> , since for these 

boundary conditions the problem is analogous to that of the 

linear harmonic oscillator (Sec. 11). Because of the rapid 



270 THE ROTATION AND VIBRATION OF MOLECULES [X-35c 

decrease in the harmonic oscillator functions outside of the 
classically permitted region (see Fig. 11-3), it does not introduce 
a serious error to consider that the two sets of boundary conditions 
are practically equivalent, so that as an approximation we may 
use the harmonic oscillator wave functions for the functions S. 

The energy levels are, therefore, using the results of Section 
11a, 

W„ K - K(K + 1), - -V<f+^-, + („ + l)w, 

(35-19) 
in which 



1 [ krl + QKjK + l)<r\ H 
2tt\ /ir» 



r 



(35-20) 



and v is the vibrational quantum number (corresponding to the 
quantum number n for the harmonic oscillator), which can take 
on the values 0, 1, 2, • • • . The functions £(f) are (Sec. 11) 

Sr(f) = {(f)^}^ _fr! ^ (V " r) ' (35_21) 

in which a — 4w 2 nv' e /h and f = p — a = r — r e — a, and H v is 
the tfth Hermite polynomial. 

The values of fc, r e , and <r for actual molecules are of such 
magnitudes that the expression for W can be considerably simpli- 
fied without loss of accuracy by the use of the expansions 

1 = 1 L _ SK(K + l )cr \ 

3K(K + l)<r + l Akr 2 } 2 kr 2 \ ' kr 2 e ~~ "^ /' 

l/^r e 2 +6K(^ + lV\ ,/2 



"° 27r\ ~^1 J 

Introducing these into Equation 35-19, we obtain for W the 
expression 

{K(K + l)v\* 



(v + 0*,. 



W 9 .k = U + £ ]A„. + #(# + l)«r 



X2* 



in which only the first terms of the expansions have been used 
and the symbol v 6 is given by 



X-35d] VIBRATION OF DIATOMIC MOLECULES 271 



1 Ik 



(35-22) 



Replacing k by its expression in terms of v, and introducing 
the value of Equation 35-17 for a, we finally obtain for W the 
expression 

IP..* = ^ + a j*'. + W + Dg^ - -i28^|7r " (35 " 23) 

The first term is evidently the vibrational energy of the mole- 
cule, considered as a harmonic oscillator. The second term is 
the energy of rotation, assuming that the molecule is a rigid 
body, 1 while the third term is the correction which takes account 
of the stretching of the actual, non-rigid molecule due to the 
rotation. The terms of higher order are unreliable because 
of the inaccuracy of the assumed potential function. 

The experimental data for most molecules fit Equation 35-23 
fairly well. For more refined work additional correction terms 
are needed, one of which will be obtained in the next section. 

35d. A More Accurate Treatment. The Morse Function. — 
The simple treatment which we have just given fails to agree 
with experiment in that it yields equally spaced levels, whereas 
the observed vibrational levels show a convergence for increasing 
values of v. In order to obtain this feature a potential function 
U(r) is required which is closer to the true U(r) described in 

1 This is seen by allowing k to become infinite, causing the third term to 
vanish (because y e — ►«>). A rigid molecule would have no vibrational 
energy, so the first term would become an additive constant. The rigid 
rotator is often discussed as a separate problem, with the wave equation 

1 d ( . dA 1 dV 8tt 2 / 

( sin t?— + ~ + — -WV =0, (35-24) 

sin#ch>\ <W/ sin 2 tfd<? 2 h 2 

the solutions of which are \j/ = &m (<p)Qkm(&), in which $ and are given by 

h 2 
Equations 35-5 and 35-6. The energy levels are Wk = K(K + 1) -• 

Sir 2 ! 

The rigid rotator is of course an idealization which does not occur in nature. 
Another idealized problem is the rigid rotator in a plane, for which the 
wave equation is 

£ + -» w * - °- (35 - 25) 

The solutions are $ = sin M*p and ^ = cos M<p, M = 0, 1, 2, • • • , and 
the energy levels are Wm = M 2 h 2 /Sw 2 T (Sec. 25a). 



272 THE ROTATION AND VIBRATION OF MOLECULES [X-35d 



Section 356, especially with regard to its behavior for large 
values of r. 

Morse 1 proposed a function of the form 

U(r) = D{1 - e-^-^} 2 , (35^26) 

which is plotted in Figure 34r-2. It has a minimum value of 
zero at r = r e and approaches a finite value D for r large. It 
therefore agrees with the qualitative considerations of Section 356 
except for its behavior at r = 0. At this point the true U(r) 
is infinite, whereas the Morse function is finite. However, the 
Morse function is very large at this point, and this deficiency 
is not a serious one. 

With the introduction of this function, the radial equation 
35-13 becomes 



« + {-«*£« + »*or-i>-i»r~~ + 



2De- 



Ur~r ) 



If we make the substitutions 

y = e -air-r.) an( J 



\s = 0. 



A = K{K + 1) 



8xV. 2 



the radial equation becomes 

a 2 n 2 \ jr y j/ 2 ry 



dy* y dy 



(35-27) 



(35-28) 



(35-29) 



The quantity r%/r 2 may be expanded in terms of y in the following 
way: 2 



0-5?)' 



= i + ~(v 

ar e 



1 >+(-k + ^- 1)f 



+ 



(35-30) 



the series being the Taylor expansion of the second expression 
in powers of (y — 1). Using the first three terms of this expan- 
sion in Equation 35-29 we obtain the result 



1 P. M. Morse, Phys. Rev. 34, 57 (1929). 

1 This treatment is due to C. L. Pekeris, Phys. Rev. 46, 98 (1934). 
solved the equation for the case K = only. 



Morse 



X-36d] 

d*S 



VIBRATION OF DIATOMIC MOLECULES 



273 



IdS + **(W^D-c. + 2D-c 1 _ D _ \ 
ydy a 2 h 2 \ y 2 V / 



in which 



C2 = A(-i-+jy. 



(35-31) 



(35-32) 



The substitutions 

S(y) = e~W(z), 
2 = 2dy, 



8 



TC'H 



«P=^H7P+«0, 



6 2 = - 



a 2 /i 2 
32*r 



(35-33) 



a 2 A 



^(TT-Z)-Co)j 



simplify Equation 35-31 considerably, yielding the equation 
dW 



dz 



.t + 



in which 



C-^ - >) 



4**n 
a 2 h 2 d 



f + '-F-O, 
dz z 



1, 



{2D - Cl ) - 5(6 + 1) 



(35-34) 



(35-35) 



Equation 35-34 is closely related to the radial equation 18-37 
of the hydrogen atom and may be solved in exactly the same 
manner. If this is done, it is found that it is necessary to 
restrict v to the values 0, 1, 2, • • • in order to obtain a poly- 
nomial solution. ' If we solve for W by means of Equations 35-35 
and the definitions of Equations 35-33, 35-32, and 35-28, we 
obtain the equation 



Wk., = D+c - 



(D + a) 



2 ah(D - V 2Cl ) 



7rVWl> + c,\ V 

oW/ , IV 



1 The solutions for v integral satisfy the boundary conditions F — ► as 
r _► _ oo instead of as r — ► (Sec. 36c). 



274 THE ROTATION AND VIBRATION OF MOLECULES [X-36c 

By expanding in terms of powers of ci/D and c 2 /Z>, this relatior 
may be brought into the form usually employed in the study ol 
observed spectra; namely, 

TT = *(" + 5) - x **l? + tj + K(K + 1)Be + 

D e K\K + l) 2 - a e (v + V 2 )K(K + 1), (35-36) 

in which c is the velocity of light, and 1 



a J2D 



Xe ~~£D' 

B e = g^, ) (35-37) 



a e = 






167rV?Z>' 



ir*D\ar e a 2 r 2 J 



For nearly all molecules this relation gives very accurate 
values for the energy levels; for a few molecules only is it neces- 
sary to consider further refinements. 

We shall not discuss the wave functions for this problem. 
They are given in the two references quoted. 

Problem 35-1. Another approximate potential function which has beer 
used for diatomic molecules 2 is 

B Ze* 

U(r) « - 

r 2 r 

Obtain the energy levels for a diatomic molecule with such a potential func- 
tion, using the polynomial method. (Hint: Follow the procedure of Sec. IS 
closely.) Expand the expression for the energy so obtained in powers o) 

07T 2 /i 

(K + l) 2 ~r~^ an d compare with Equation 35-23. Also obtain the positioi 
h 2 B 

of the minimum of U(r) and the curvature of U(r) at the minimum. 

Problem 36-2. Solve Equation 35-35 for the energy levels. 

1 The symbol o>« is often used in place of £«. 

2 E. Ftjes, Ann. d. Phys. 80, 367 (1926). 



X-86a] THE ROTATION OF POLYATOMIC MOLECULES 275 

36. THE ROTATION OF POLYATOMIC MOLECULES 

The straightforward way to treat the rotational and vibrational 
motion of a polyatomic molecule would be to set up the wave 
equation for ^ ntV (£) (Eq. 34-4), introducing for U n (i) an expres- 
sion obtained either by solution of the electronic wave equation 
34-3 or by some empirical method, and then to solve this nuclear 
wave equation, using some approximation method if necessary. 
This treatment, however, has proved to be so difficult that it is 
customary to begin by making the approximation of neglecting 
all interaction between the rotational motion and the vibrational 
motion of the molecule. 1 The nuclear wave equation can then 
be separated into two equations, one, called the rotational wave 
equation, representing the rotational motion of a rigid body. 
In the following paragraphs we shall discuss this equation, 
first for the special case of the so-called symmetrical-top molecules, 
for which two of the principal moments of inertia are equal 
(Sec. 36a), and then for the unsymmetrical-top molecules, for 
which the three principal moments of inertia are unequal (Sec. 
366). The second of the two equations into which the nuclear 
wave equation is separated is the vibrational wave equation, 
representing the vibrational motion of the non-rotating molecule. 
This equation will be treated in Section 37, with the usual 
simplifying assumption of Hooke's-law forces, the potential 
energy being expressed as a quadratic function of the nuclear 
coordinates. 

36a. The Rotation of Symmetrical-top Molecules. — A rigid 
body in which two of the three principal moments of inertia 2 

iSee, however, C. Eckart, Phys. Rev. 47, 552 (1935); J. H. Van Vleck, 
ibid. 47, 487 (1935); D. M. Dennison and M. Johnson, ibid. 47, 93 (1935). 

2 Every body has three axes the use of which permits the kinetic energy 
to be expressed in a particularly simple form. These are called the principal 
axes of inertia. The moment of inertia about a principal axis is denned by 
the expression Jpr 2 dr, in which p is the density of matter in a given volume 
element dr, r is the perpendicular distance of this element from the axis iii 
question, and the integration is over the entire volume of the solid. For a. 
discussion of this question see J. C. Slater and N. II. Frank, " Introduction 
to Theoretical Physics," p. 94, McGraw-Hill Book Company, Inc., New 
York, 1933. 

In case that a molecule possesses an n-fold symmetry axis with n greater 
than 2 (such as ammonia, with a three-fold axis), then two principal moments 
of inertia about axes perpendicular to this symmetry axis are equal, and the 



276 THE ROTATION AND VIBRATION OF MOLECULES [X-36a 

are equal is called a symmetrical top. Its position in space is 
best described by the use of the three Eulerian angles #, «*>, and x 
shown in Figure 36-1. # and v are the ordinary polar-coordinate 
angles of the axis of the top while x (usually called \p) is the 
angle measuring the rotation about this axis. 

Since we have considered only assemblages of point particles 
heretofore, we have not given the rules for setting up the wave 
equation for a rigid body. We shall not discuss these rules 
here 1 but shall take the wave equation for the symmetrical top 




Fia. 36-1. — Diagram showing Eulerian angles. 

from the work of others. 2 Using C to represent the moment of 
inertia about the symmetry axis and A the two other equal 
moments of inertia, this wave equation is 



molecule is a symmetrical top. A two-fold axis does not produce a symmet- 
rical-top molecule (example, water). If the molecule possesses two or more 
symmetry axes with n greater than 2, it is called a spherical-top molecule, 
all three moments of inertia being equal. 

1 Since the dynamics of rigid bodies is based on the dynamics of particles, 
these rules must be related to the rules given in Chapter IV. For a dis- 
cussion of a method of finding the wave equation for a system whose Hamil- 
tonian is not expressed in Cartesian coordinates, see B. Podolsky, Phys. 
Rev. 32, 812 (1928), and for the specific application to the symmetrical 
top see the references below. 

2 F. Reiche and H. Rademacher, Z. f. Phys. 39, 444 (1926); 41, 453 
(1927); R. de L. Kroniq and I. I. Rabi, Phys. Rev. 29, 262 (1927). D. M. 
Dennison, Phys. Rev. 28, 318 (1926), was the first to obtain the energy 
levels for this system, using matrix mechanics rather than wave mechanics. 



X-36a) THE ROTATION OF POLYATOMIC MOLECULES 277 
+ 



K*"8) 



1 

sin 2 




/cos' 
\sin 


2 # 
2 # 


< 






- 


2cos# 
sin 2 # 


ay 


+ 


Br* AW, 






sin#d#\ 

(36-1) 

The angles x and <p do not occur in this equation, although 
derivatives with respect to them do. They are therefore cyclic 
coordinates (Sec. 17), and we know that they enter the wave 
function in the following manner: 

^ = e(d)e iM *e iK x, (36-2) 

in which M and K have the integral values 0, ±1, ±2, • * • . 
Substitution of this expression in the wave equation confirms 
this, yielding as the equation in # 

1 d ( . d&\ j_M^ , /cos 2 * JA 2 
Sr7# ^V Sm 3*/ ~ W 2 * + Un 2 t> + Cr 

- 2^W - *£w)e = 0. (3(W) 

sm 2 # h 2 ) 

We see that # = and # = 7r are singular points for this equation 
(Sec. 17). It is convenient to eliminate the trigonometric 
functions by the change of variables 



x = y 2 {\ - cos #U 

e(*) = r(x), I 

at the same time introducing the abbreviation 



(36-4) 



x = S^W _ A^ 2> (36 _ 5) 

the result being 

U* - *>£} + i x - "t$*:> 1)|, > - ° <-» 

The singular points, which are regular points, have now been 
shifted to the points and 1 of x, so that the indicial equation 
must be obtained at each of these points. Making the sub- 
stitution T(x) = x a G(x), we find by the procedure of Section 17 
that s equals Vi\K — M\, while the substitution 

T(x) = (1 - xYH{\ - x) 



278 THE ROTATION AND VIBRATION OF MOLECULES [X-36a 

yields a value of }4\K + M\ for s'. Following the method of 
Section 18c we therefore make the substitution 

0(0) = T(x) = xW K ~ Ml (l - *)W*+*iF(a;), (36-7) 

which leads to the equation 1 for F 

x(l - x)g + (a - 0x)^ + 7^=0, (36-8) 

in which 

a = \K - M\ + 1, 

P = \K + M\ + \K- M\+2. 

and 

7 = X + JC* - (Ml* + m\+ y 2 \K - m\kv 2 \k + M\ + 

y 2 \K -m\ + i). 

We can now apply the polynomial method to this equation by 
substituting the series expression 



F(x) = ^a„z" 



in Equation 36-8. The recursion formula which results is 

For this to break off after the jth term (the series is not an 
acceptable wave function unless it terminates), it is necessary 
for the numerator of Equation 36-9 to vanish, a condition which 
leads to the equation for the energy levels 

in which 

j = j + y 2 \K + M\ + y 2 \K - M\ t (36-11) 

that is, J is equal to, or larger than, the larger of the two quan- 
tities \K\ and \M\. The quantum number J is therefore zero or a 
positive integer, so that we have as the allowed values of the 
three quantum numbers 

J = 0, 1, 2, • • • , ) 

K = 0, ±1, ±2, • • • , ±J\ (36-12) 

M =0, ±1, ±2, • • • , ±J.) 

1 This equation is well known to mathematicians as the hypergeomelric 
equation. 



X-S6a] THE ROTATION OF POLYATOMIC MOLECULES 



279 



h 2 
It can be shown that J(J + l)l~2 k *ke s Q^are of the total 

angular momentum, while Kh/2ir is the component of angular 
momentum along the symmetry axis of the top and Mh/2v the 
component along an arbitrary axis fixed in space. 



—55 



Wt 



•54 



— 55 



-53 



•44 



•52 



-51 



-43 



50 



-4E 



-33,41 



-32 



21 



•22 



-21 



30 



20 



A*2C 



-00 



•54 



-53 



-52 r 



S.50 



-44 



•43 
—42 



41 



40 



-33 



-32 



,31 



30 



-22 



Ji-10 



A-iC 



-00 



Fio. 3&-2. — Energy-level diagram for symmetrical-top molecule, with A = 2C 
and with ii » J^C. Values of the quantum numbers / and K are given for 
each level. 

When K is zero, the expression for W reduces to that for the 
simple rotator in space, given in a footnote in Section 35c. The 
energy does not depend on M or on the sign of K, and hence 
the degeneracy of a level with given J is 2 J + 1 or 4 J -+- 2, depend- 
ing on whether K is equal to zero or not. The appearance 1 of 
the set of energy levels depends on the relative magnitudes of 
A and C, as shown in Figure 3&-2. 

1 For a discussion of the nature of these energy levels and of the spectral 
lines arising from them, see D. M. Dennison, Rev. Mod. Phy$. 3, 280 (1931). 



280 THE ROTATION AND VIBRATION OF MOLECULES [X-8«b 

The wave functions can be constructed by the use of the 
recursion formula 36-9. In terms of the hypergeometric 
functions 1 F(a, b; c;x), the wave function is 

+jkm(#, <p, x) = N JKU x^ K - m {l - a.)W*+*l e «*H-*x> 

F(-J + MP - 1,J + MP;1 + \K - M\; x), (36-13) 

in which x = \i{\ — cos d) and 

v _ ( _ ( 2J+l)(J+y 2 \K+M\+V 2 \K-M\)\ 
^">>-\Sw*(J-yi\K+M\-H\K-M\)\(\K-M\\)* 

(J-y^K+Ml+mK-MWX* 
(J + y 2 \K+M\-y 2 \K-M\)\) ^°- iV 

In case that all three principal moments of inertia of a molecule 
are equal, the molecule is called a spherical-top molecule (examples: 
methane, carbon tetrachloride, sulfur hexafluoride). The energy- 
levels in this case assume a particularly simple form (Problem 
36-2). 

It has been found possible to discuss the rotational motion of 
molecules containing parts capable of free rotation relative to 
other parts of the molecule. Nielsen 2 ha? treated the ethane 
molecule, assuming the two methyl groups to rotate freely 
relative to one another about the C-C axis, and La Coste 3 has 
similarly discussed the tetramethylmethane molecule, assuming 
free rotation of each of the four methyl groups about the axis 
connecting it with the central carbon atom. 

Problem 36-1. Using Equation 36-9, construct the polynomial F(x) 
for the first few sets of quantum numbers. 

Problem 36-2. Set up the expression for the rotational energy levels for 
a spherical-top molecule, and discuss the degeneracy of the levels. Calcu- 
late the term values for the six lowest levels for the methane molecule, 
assuming the C-H distance to be 1 .06 A. 

36b. The Rotation of Unsymmetrical-top Molecules. — The 

treatment of the rotational motion of a molecule with all three 
principal moments of inertia different (called an unsymmetrical- 
top molecule) is a much more difficult problem than that of the 
preceding section for the symmetrical top. We shall outline a 

1 The hypergeometric function is discussed in Whittaker and Watson, 
"Modern Analysis," Chap. XIV. 

" H. H. Nielsen, Phys. Rev. 40, 445 (1932). 
*L. J. B. La Coste. Phys. Rev. 46. 718 (1934). 



X-36b] THE ROTATION OF POLYATOMIC MOLECULES 281 

procedure which has been used with success in the interpretation 
of the spectra of molecules of this type. 

Let us write the wave equation symbolically as 

Hyp = W$. (36-15) 

Inasmuch as the known solutions of the wave equation for a 
symmetrical-top molecule form a complete set of orthogonal 
functions (discussed in the preceding section), we can expand 
the wave function \p in terms of them, writing 

* = % a JKM r JKM , (36-16) 

JKM 

in which we use the symbol \p] KM to represent the symmetrical-top 
wave functions for a hypothetical molecule with moments of 
inertia A 0y B (= A ), and C . If we now set up the secular 
equation corresponding to the use of the series of Equation 36-16 
as a solution of the unsymmetrical-top wave equation (Sec. 27a), 
we find that the only integrals which are not zero are those 
between functions with the same values of J and M y so that 
the secular equation immediately factors into equations corre- 
sponding to variation functions of the type 1 

4- J 

ypjoM = 2) a JK*tfjKM' (36-17) 

On substituting this expression in the wave equation 36-15, 
we obtain the equation 

%a K H+° K = WX*** * ( 36 ~ 18 ) 

K K 

in which for simplicity we have omitted the subscripts J and M, 
the argument from now on being understood to refer to definite 
values of these two quantum numbers. On multiplication by 
yf/\* and integration, this equation leads to the following set of 
simultaneous homogeneous linear equations in the coefficients a K : 



s 



a K (H L K-tLKW)=0, L=-J,-J+l, •••,+/, (36-19) 



in which 8lk has the value 1 f or L = K and otherwise, and H L k 

1 The same result follows from the observation that J and M correspond 
to the total angular momentum of the system and its component along a 
fixed axis in space (see Sec. 52, Chap. XV). 



282 THE ROTATION AND VIBRATION OF MOLECULES [X-87a 

represents the integral JVi *£ty£dr. This set of equations has a 
solution only for values of W satisfying the secular equation 

H-j,-j — W H-j,-j+i 



H- Jt j 
H-j+i,j 



Hj,j - W 



= 0. 

(36-20) 



These values of W are then the allowed values for the rotational 
energy of the unsymmetrical-top molecule. Wang 1 has evaluated 
the integrals Hlk and shown that the secular equation can 
be further simplified. The application in the interpretation 
of the rotational fine structure of spectra has been carried 
out in several cases, including water, 2 hydrogen sulfide, 3 and 
formaldehyde. 4 

37. THE VIBRATION OF POLYATOMIC MOLECULES 

The vibrational motion of polyatomic molecules is usually 
treated with an accuracy equivalent to that of the simple dis- 
cussion of diatomic molecules given in Section 35c, that is, with 
the assumption of Hooke's-law forces between the atoms. When 
greater accuracy is needed, perturbation methods are employed. 

Having made the assumption of Hooke's-law forces, we employ 
the method of normal coordinates to reduce the problem to soluble 
form. This method is applicable whether we use classical 
mechanics or quantum mechanics. Inasmuch as the former 
provides a simpler introduction to the method, we shall consider 
it first. 

37a. Normal Coordinates in Classical Mechanics. — Let the 
positions of the n nuclei in the molecule be described by giving 
the Cartesian coordinates of each nucleus referred to the 
equilibrium position of that nucleus as origin, as shown in Figure 
37-1. Let us call these coordinates q[ } q' 2r • • • , q' Zn . In terms 
of them we may write the kinetic energy of the molecule in the 
form 

1 S. C. Wang, Phys. Rev. 34, 243 (1929). See also H. A. Kramers and 
G. P. Ittmann, Z. f. Phys. 53, 553 (1929); 68, 217 (1929); 60, 663 (1930); 
O. Klein, Z. f. Phys. 58, 730 (1929); E. E. Witmer, Proc. Not. Acad. Sci. 
13, 60 (1927); H. H. Nielsen, Phys. Rev. 38, 1432 (1931). 

> R. Mecke, Z. f. Phys. 81, 313 (1933). 

8 P. C. Cross, Phys. Rev. 47, 7 (1935). 

4 G. H. Dieke and G. B. Kistiakowskt, Phys. Rev. 46, 4 (1934). 



X-S7a] THE VIBRATION OF POLYATOMIC MOLECULES 

3» 



T-H%M&; 



283 
(37-1) 



t-1 



in which M »• is the mass of the nucleus with coordinate q[. By 
changing the scale of the coordinates by means of the relation 

qi = y/Mrfi, i = 1, 2, • • • , 3n, (37-2) 

we can eliminate the masses from the kinetic energy expression, 
obtaining 

T = yi%& (37-3) 

The potential energy V depends on the mutual positions of 
the nuclei and therefore upon the coordinates g». If we restrict 



Origin 



-*-2 



6 






Origin 



9 



Origin 



# *7 ^6 *9 
— >8 



Fia. 37-1. — Coordinates q[ . . . q** of atoms measured relative to equilibrium 

positions. 

ourselves to a discussion of small vibrations, we may expand 
V as a Taylor series in powers of the g's, 



Vfatf* 



<7s») = V + 2^j «< + gS 6 '**' + " " ' ' 



(87-4) 



284 THE ROTATION AND VIBRATION OF MOLECULES [X-37a 
in which &,-,• is given by 

ba 



bii ~ \dlidqJo' 



and the subscript means that the derivatives are evaluated 
at the point q\ = 0, g 2 = 0, etc. If we choose our zero of energy 
so that V equals zero when q h q 2j etc. are zero, then V is zero. 
Likewise the second term is zero, because by our choice of coordi- 
nate axes the equilibrium position is the configuration qi = 0, 
q 2 = 0, etc., and the condition for equilibrium is 



(sa-* 



i = 1, 2, • • • , 3n. (37-5) 

Neglecting higher terms, we therefore write 

Viqiq* ' ' ' <Z3„) = MtyM,. (37-6) 

v 

Using the coordinates q^ we now set up the classical equations 
of motion in the Lagrangian form (Sec. lc). In this case the 
kinetic energy T is a function of the velocities qi only, and the 
potential energy V is a function of the coordinates qi only, and 
in consequence the Lagrangian equations have the form 

On introducing the above expressions for T and V we obtain the 
equations of motion 

9k + %b ikqi = 0, k = 1, 2, • • • , 3n. (37-8) 

In case that the potential-energy function involves only squares 
q% and no cross-products q t gj with i 5* j; that is, if ba vanishes 
for i 9^ j, then these equations of motion can be solved at once. 
They have the form 

qu + b kk q k =0, k = 1, 2, • • • , 3n, (37-9) 

the solutions of which are (Sec. la) 

9* = q$ sin (\/Sirf + «*)i * = If 2, • • • , 3n, (37-10) 

in which the gj's are amplitude constants and the 8 k s phase 
constants of integration. In this special case, then, each of the 



X-37aJ THE VIBRATION OF POLYATOMIC MOLECULES 285 

coordinates q k undergoes harmonic oscillation, the frequency 
being determined by the constant b kk . 

Now it is always possible by a simple transformation of 
variables to change the equations of motion from the form 
37-8 to the form 37-9; that is, to eliminate the cross-products 
from the potential energy and at the same time retain the form 
37-3 for the kinetic energy. Let us call these new coordinates 
Qi(l = 1, 2, • • • , 3n). In terms of them the kinetic and the 
potential energy would be written 

T = y 2 %Q? (37-11) 

i 
and 

V = V 2 %\iQl (37-12) 

i 

and the solutions of the equations of motion would be 

Qi = QJsin (VXit + 8i), I = 1, 2, • • • , 3n. (37-13) 

Instead of finding the equations of transformation from the 
q's to the Q's by the consideration of the kinetic and potential 
energy functions, we shall make use of the equations of motion. 
In case that all of the amplitude constants Q$ are zero except one, 
Qi; sa y> then Qi will vary with the time in accordance with 
Equation 37-13, and, inasmuch as the q's are related to the 
Q's by the linear relation 

3n 
qk = X BklQl > (37 " 14) 

each of the q's will vary with the time in the same way, namely, 
q k = A k sin (V\t + 5 X ), k = 1, 2, • • • , 3n. (37-15) 

In these equations A k represents the product B k iQ[, and X the 
quantity Xi, inasmuch as we selected Q\ as the excited coordinate; 
the new symbols are introduced for generality. On substituting 
these expressions in the equations of motion 37-8, we obtain the 
set of equations 

3n 

-\A k + %b ik Ai = 0, k = 1,2, • • • , 3n. (37-16) 

This is a set of 3n simultaneous linear homogeneous equations 
in the 3n unknown quantities A k . As we know well by this time 



286 THE ROTATION AND VIBRATION OF MOLECULES [X-37a 

(after Sees. 24, 26d, etc.), this set of equations possesses a solution 
other than the trivial one A x = A 2 = # * • =0 only when the 
corresponding determinantal equation (the secular equation of 
perturbation and variation problems) is satisfied. This equa- 
tion is 

6n — X 612 • • • biBn 

b *\ &22 7 x ; ; ; bi * = o. (37-17) 

bznl Ozn2 ' ' * #8n8n ~~ X 

In other words, Equation 37-15 can represent a solution of the 
equations of motion only when X has one of the 3n values which 
satisfy Equation 37-17. (Some of these roots may be equal.) 
Having found one of these roots, we can substitute it in Equation 
37-16 and solve for the ratios 1 of the A's. If we put 

A k i = B kl Ql (37-18) 

and introduce the extra condition 

X B it = l > (37-19) 

k 

in which the subscript I specifies which root \i of the secular 
equation has been used, then we can determine the values of 
the Bus, Q\ being left arbitrary. 

By this procedure we have obtained 3n particular solutions 
of the equations of motion, one for each root of the secular 
equation. A general solution may be obtained by adding all 
of these together, a process which yields the equations 

3n 

q k = X Q ° Bkl sin (v ^' + 5z) - (37 ~ 20) 

This solution of the equations of motion contains 6n arbitrary 
constants, the amplitudes Q° t and the phases 6i f which in 
any particular case are determined from a knowledge of the 
initial positions and velocities of the n nuclei. 

We have thus solved the classical problem of determining the 
positions of the nuclei as a function of the time, given any set 
of initial conditions. Let us now discuss the nature of the 

1 These equations are homogeneous, so that only the ratios of the A's can 
be determined. The extra condition 37-19 on the Bus then allows them 
to be completely determined. 



X-37a] THE VIBRATION OF POLYATOMIC MOLECULES 287 

solution. As mentioned above, if we start the molecule vibrating 
in such a way that all the QJ's except one, say Q° u are zero, the 
solution is 

q k = Q\B kl sin (y/\ x t + h x ), k = 1, 2, • • • , 3n, (37-21) 

which shows that each of the nuclei carries out a simple harmonic 
oscillation about its equilibrium position with the frequency 

v x = ^. (37-22) 

All of the nuclei move with the same frequency and the same 
phase; that is, they all pass through their equilibrium positions 
at the same time and reach 
their positions of maximum 
amplitude at the same time. 
These amplitudes, however, 
are not the same for the differ- 
ent nuclei but depend on the 
values of the B k i's and upon the 
initial amplitude, which is 
determined by Q?. A vibration 
governed by Equation 37-21 
and therefore having these prop- 
erties is called a normal mode Fig. 37-2.™One of the normal modes 

of vibration of a symmetrical triatomic 

of Vibration of the System (see molecule. Each of the atoms moves in 

Fiff 37—2^ an( * out a ^ ong a r& dial direction as shown 

®* '" .ji by the arrows. All the atoms move with 

It is not required, however, the same frequency and phase, and in this 

that the nuclei have initial special case with the same amplitude. 

amplitudes and velocities such that the molecule undergo such a 
special motion. We can start the molecule off in any desired man- 
ner, with the general result that many of the constants Q\ will be 
different from zero. In such a case the subsequent motion of 
the nuclei may be thought of as corresponding to a superposition 
of normal vibrations, each with its own frequency V%/27r and 
amplitude Q?. The actual motion may be very complicated, 
although the normal modes of vibration themselves are fre- 
quently quite simple. 

The normal coordinates of the system are the coordinates Q h 
which we introduced in Equation 37-14. These coordinates 
specify the configuration of the system just as definitely as the 
original coordinates q%. 




288 THE ROTATION AND VIBRATION OF MOLECULES [X-37b 

The expansion of V given in Equation 37-4 is not valid except 
when the nuclei stay near their equilibrium positions. That is, 
we have assumed that the molecule is not undergoing transla- 
tional or rotational motion as a whole. Closely related to this 
is the fact, which we shall not prove, that zero occurs six 1 times 
among the roots X* of the secular equation. The six normal 
modes of motion corresponding to these roots, which are not 
modes of vibration because they have zero frequency, are the 
three motions of translation in the x ) y, and z directions and 
the three motions of rotation about the x } y ) and z axes. 

37b. Normal Coordinates in Quantum Mechanics. — It can be 
shown 2 that when the coefficients B k i of Equation 37-14 are 
determined in the manner described in the last section, the 
introduction of the transformation 37-14 for the g^'s into the 
expression for the potential energy yields the result 

V = KX b »M> = ^2 XlQ ?; (37-23) 

that is, the transformation to normal coordinates has eliminated 
the cross-products from the expression for the potential energy. 
In addition, this transformation has the property of leaving the 
expression for the kinetic energy unchanged in form; 3 i.e., 

T = «2« - «SQf. (37 _ 24 ) 

These properties of the normal coordinates enable us to treat the 
problem of the vibrations of polyatomic molecules by the 
methods of quantum mechanics. 

The wave equation for the nuclear motion of a molecule is 



2i v ^ + *w {w ~ F) * = °> (37 ~ 25) 

i-i 
in which ^ represents the nuclear wave function ^ n ,*(£) of Equa- 

1 This becomes five for linear molecules, which have only two degrees of 
rotational freedom. 

2 For a proof of this see E. T. Whittaker, "Analytical Dynamics," 
Sec. 77, Cambridge University Press, 1927. 

3 A transformation which leaves a simple sum of squares unaltered is 
called an orthogonal transformation. 



X-37bl THE VIBRATION OF POLYATOMIC MOLECULES 289 

tion 34-4. In terms of the Cartesian coordinates g- previously 
described (Fig. 37-1), we write 

n 3n 

; - 1 t - 1 

By changing the scale of the coordinates as indicated by Equation 
37-2 we eliminate the APs, obtaining for the wave equation the 
expression 

3n 

2fS+^V-F)* = 0. (37-27) 

1 = 1 

We now introduce the normal coordinates Qi. The reader can 
easily convince himself that an orthogonal transformation will 
leave the form of the first sum in the wave equation unaltered, 
so that, using also Equation 37-23, we obtain the wave equation 
in the form 

3n 3n 

1=1 /=1 

This equation, however, is immediately separable into 3n 
one-dimensional equations. We put 

* = MQMQt) • • • lMQ«»), (37-29) 

and obtain the equations 

m+^ w > -¥**)*> -°> (37 " 30) 

each of which is identical with the equation for the one-dimen- 
sional harmonic oscillator (Sec. lla). The total energy W is 
the sum of the energies Wk associated with each normal coordi- 
nate; that is, 

3n 

W = 2jTT*. (37-31) 

The energy levels of the harmonic oscillator were found in 
Section lla to have the values (v + M)hv , where v is the quan- 
tum number and v the classical frequency of the oscillator. 
Applying this to the problem of the polyatomic molecule, we 
see that 



290 THE ROTATION AND VIBRATION OF MOLECULES [X-38 

W = %W k = %(v k + V 2 )hv k} (37 _ 3 2) 

jfc k 

in which v k is the quantum number (v k = 0, 1, 2, • • • ) and v k \a 
the classical frequency of the kth normal mode of vibration. 
We have already seen (from Eq. 37-22) that 

v k = ^- (37-33) 

The energy-level diagram of a polyatomic molecule is therefore 
quite complex. If, however, we consider only the fundamental 
frequencies emitted or absorbed by such a molecule; that is, the 
frequencies due to a change of only one quantum number v k 
by one unit, we see that these frequencies are v u v 2 , • • • , vz n ) 
that is, they are the classical frequencies of motion of the 
molecule. 

This type of treatment has been very useful as a basis for the 
interpretation of the vibrational spectra of polyatomic molecules. 
Symmetry considerations have been widely employed to simplify 
the solution of the secular equation and in that connection the 
branch of mathematics known as group theory has been very 
helpful. 1 

38. THE ROTATION OF MOLECULES IN CRYSTALS 

In the previous sections we have discussed the rotation and 
vibration of free molecules, that is, of molecules in the gas phase. 
There is strong evidence 2 that molecules and parts of molecules 
in many crystals can rotate if the temperature is sufficiently 
high. The application 23 of quantum mechanics to this problem 
has led to a clarification of the nature of the motion of a molecule 
within a crystal which is of some interest. The problem is 
closely related to that dealing with the rotation of one part of a 
molecule relative to the other parts, such as the rotation of 
methyl groups in hydrocarbon molecules. 4 

1 C. J. Brbster, Z. f. Phys. 24, 324 (1924); E. Wigner, Gottinger Nachr. 
133 (1930); G. Placzek, Z. f. Phys. 70, 84 (1931); E. B. Wilson, Jr., 
Phys. Rev. 46, 706 (1934); /. Chem. Phys. 2, 432 (1934); and others. 

*L. Pauling, Phys. Rev. 36, 430 (1930). This paper discusses the 
mathematics of the plane rotator in a crystal as well as the empirical evidence 
for rotation. 

8 T. E. Stern, Proc. Roy. Soc. A 130, 551 (1931). 

4 E. Teller and K. Weigert, Gdttinger Nachr. 218 (1933); J. E. 
Lennard-Jones and H. H. M. Pike, Trans. Faraday Soc. 30, 830 (1934). 



X-38] THE ROTATION OF MOLECULES IN CRYSTALS 291 



The wave equation for a diatomic molecule in a crystal, con- 
sidered as a rigid rotator, obtained by introducing V into the 
equation for the free rotator given in a footnote of Section 35c, is 



1 

sin# 



£(-3) 



+ 



l av 



+^v 



V)+ = 0, (38-1) 



sin 2 # dip 2 

in which # and <p are the polar coordinates of the axis and J is 
the moment of inertia of the molecule. The potential function V 
is introduced as an approximate description of the effects of the 
other molecules of the crystal upon the molecule in question. 



2V - 




Fio. 38-1. — Idealized potential function for a symmetrical diatomic molecule in 

a crystal. 

If the molecule being studied is made up of like atoms, such as 
is O2 or H 2 , then a reasonable form to assume for V is 

V = 7 (1 - cos 20), (38-2) 

which is shown in Figure 38-1. Turning a symmetrical molecule 
end for end does not change 7, as is shown in the figure by the 
periodicity of V with period w. 

The wave equation 38-1 with the above form for V has been 
studied by Stern, 1 who used the mathematical treatment given 
by A. H. Wilson. 2 We shall not reproduce their work, although 
the method of solution is of some interest. The first steps are 
exactly the same as in the solution of the equation discussed in 
Section 18c except that a three-term recursion formula is obtained. 
The method of obtaining the energy levels from this three-term 
formula is then similar to the one which is discussed in Section 
42c, where a similar situation is encountered. 

We have referred to the case of free rotation of methyl groups at the end of 
Section 36a. 

1 T. E. Stern, loc. cit. 

1 A. H. Wilson, Proc. Roy. Soc. A 118, 628 (1928). 



292 THE ROTATION AND VIBRATION OF MOLECULES [X-38 

The results obtained may best be described by starting with 
the two limiting cases. When the energy of the molecule is 
small compared with F (i.e., at low temperatures), then the 
potential function can be regarded as parabolic in the neighbor- 
hood of the minima and we expect, as is actually found, that the 
energy levels will be those of a two-dimensional harmonic 
oscillator and that the wave functions will show that the molecule 
oscillates about either one of the two positions of equilibrium, 
with little tendency to turn end over end. When the molecule 
is in a state with energy large compared with V (i.e., at high 
temperatures), the wave functions and energy levels approximate 
those of the free rotator (Sec. 35c, footnote), the end-over-end 
motion being only slightly influenced by the potential energy. 
In the intermediate region, the quantum-mechanical treatment 
shows that there is a fairly sharp but nevertheless continuous 
transition between oscillation and rotation. In other words, for 
a given energy there is a definite probability of turning end over 
end, in sharp contrast with the results of classical mechanics, 
which are that the molecule either has enough energy to rotate 
or only enough to oscillate. 

The transition between rotation and oscillation takes place 
roughly at the temperature T = 2Vo/kj where k is Boltzmann's 
constant. This temperature lies below the melting point for a 
number of crystals, such as hydrogen chloride, methane, 
and the ammonium halides, and is recognizable experimentally 
as a transition point in the heat-capacity curve. For solid 
hydrogen even the lowest energy level is in the rotational region, 
a fact which is" of considerable significance in the application of 
the third law of thermodynamics. 

Problem 38-1. Considering the above system as a perturbed rigid rota- 
tor, study the splitting of the rotator levels by the field, indicating by an 
energy-level diagram the way in which the components of the rotator levels 
begin to change as the perturbation is increased. 

General References 

A. E. Ruark and H. C. Urey: "Atoms, Molecules and Quanta," Chap. 
XII, McGraw-Hill Book Company, Inc., New York, 1930. A general 
discussion of the types of molecular spectra found experimentally, with the 
theoretical treatment of some of them. 

W. Wbizel: " Bandenspektren, " Handbuch der Experimentalphysik 
(Wien-Harms), Erganzungsband I. A complete discussion of the theory 



X-38] THE ROTATION OF MOLECULES IN CRYSTALS 293 

and results for diatomic molecules, with some reference to polyatomic 
molecules. 

D. M. Dennison: Rev. Mod. Phys. 3, 280 (1931). A discussion of the 
methods of treating the rotational and vibrational spectra of polyatomic 
molecules. 

R. de L. Kronig: "Band Spectra and Molecular Structure,' ' Cambridge 
University Press, 1930. The theory of diatomic spectra, including con- 
sideration of electron spins. 

A. Schaefer and F. Matossi: "Das Ultrarote Spektrum," Julius 
Springer, Berlin, 1930. Methods, theory, and results of infrared 
spectroscopy. 

K. W. F. Kohlrausch : "Der Smekal-Raman-Effekt," Julius Springer, 
Berlin, 1931. 

G. Placzek: " Rayleigh-Streuung und Ramaneffekfc," Handbuch der 
Radiologic, Vol.' VI, Akademische Verlagsgesellschaft, Leipzig, 1934. 

E. Teller: "Theorie der langwelligen Molekulspektren," Hand- und 
Jahrbuch der chemisehen Physik, Vol. IX, Akademische Verlagsgesellschaft, 
Leipzig, 1934. 



CHAPTER XI 

PERTURBATION THEORY INVOLVING THE TIME, THE 

EMISSION AND ABSORPTION OF RADIATION, AND 

THE RESONANCE PHENOMENON 

39. THE TREATMENT OF A TIME-DEPENDENT PERTURBATION 
BY THE METHOD OF VARIATION OF CONSTANTS 

There have been developed two essentially different wave- 
mechanical perturbation theories. The first of these, due to 
Schrodinger, provides an approximate method of calculating 
energy values and wave functions for the stationary states of a 
system under the influence of a constant (time-independent) 
perturbation. We have discussed this theory in Chapter VI. 
The second perturbation theory, which we shall -treat in the 
following paragraphs, deals with the time behavior of a system 
under the influence of a perturbation; it permits us to discuss such 
questions as the probability of transition of the system from one 
unperturbed stationary state to another as the result of the 
perturbation. (In Section 40 we shall apply the theory to 
the problem of the emission and absorption of radiation.) The 
theory was developed by Dirac. 1 It is often called the theory 
of the variation of constants; the reason for this name will be 
evident from the following discussion. 

Let us consider an unperturbed system with wave equation 
including the time 

*•*•--£ IT (39 - 1} 

the normalized general solution of which is 

oo 

M*> = %a n *l, (39-2) 

n-0 

*P. A. M. Dirac, Proc. Roy. Soc. A 112, 661 (1926); A 114, 243 (1927). 
Less general discussions were also given by Schrddinger in his fourth 1926 
paper and by J. C. Slater, Proc. Nat. Acad. Set. 13, 7 (1927). 

294 



XI-39] TIME-DEPENDENT PERTURBATION 295 

in which the a n 's are constants, with ^a*a n = 1, and the 

n 

y„8 are the time-dependent wave functions for the stationary 
states, the corresponding energy values being Wo, W\, • • • , 
Wl, • • • . Now let us assume that the Hamiltonian for the 
actual system contains in addition to H° (which is independent 
of t) a perturbing term H r , which may be a function of the time 
as well as of the coordinates of the system. 1 (For example, 
H' might be zero except during the period t x < t < h, the 
perturbation then being effective only during this period.) 
Since we desire to express our results in terms of the unperturbed 
wave functions including the time, we must consider the Schrod- 
inger time equation for the system. This equation is 

(flf . + IT)* --£.** (39-3) 

A wave function satisfying this equation is a function of the 
time and of the coordinates of the system. For a given value 
of t, say t' , y?(t f ) is a function of the coordinates alone. By 
the general expansion theorem of Section 22 it can be represented 
as a series involving the complete set of orthogonal wave functions 
for the unperturbed system, 

*(x u • • • , z N , t') = 2 a »*S(*i> ''',*», O, (39-4) 

n 

the symbol ¥°(zi, • • • , Zn, t f ) indicating that t f is introduced 
in place of t in the exponential time factors. The quantities 
a n are constants. For any other value of t a similar expansion 
can be made, involving different values of the constants a n . 
A general solution of the wave equation 39-3 can accordingly 
be written as 

¥(*!, • • • , z N , t) = 2a n (0*2(*i, ••-,**, 0, (39-5) 

n 

the quantities a n (t) being functions of t alone, such as to cause SP 
to satisfy the wave equation 39-3. 

The nature of these functions is found by substituting the 
expression 39-5 in the wave equation 39-3, which gives 

1 W might also be a function of the momenta p xlt • • *• , which would then 

be replaced by ■— — , .... 
2irt dxi 



296 PERTURBATION THEORY INVOLVING THE TIME [XI -39a 

n n 

The first and last terms in this equation cancel (by Eqs. 39-1 and 
39-2), leaving 

n n 

If we now multiply by ^* and integrate over configuration space, 
noting that all terms on the left vanish except that for n = m 
because of the orthogonality properties of the wave functions, 
we obtain 



oo 

d m {t) = -^^a n (t)f* Q m *H'*°d Ty 



m = 0, 1, 2, 



(39-6) 

This is a set of simultaneous differential equations in the functions 
a,m{t), by means of which these functions can be evaluated in 
particular cases. 

39a. A Simple Example. — As an illustration of the use of the 
set of equations 39-6, let us consider that at the time t = 
we know that a system in which we are interested is in a particular 
stationary state, our knowledge perhaps having been obtained 
by a measurement of the energy of the system. The wave func- 
tion representing the system is then ^°, in which I has a particular 
value. If a small perturbation H ' acts on the system for a short 
time t f , H r being independent of t during this period, we may 
solve the equations 39-6 by neglecting all terms on the right side 
except that with n = I) that is, by assuming that only the term 
in ai(t) need be retained on the right side of these equations. 
It is first necessary for us to discuss the equation for aj itself. 
This equation (Eq. 39-6 with m = I and a n = for n 5* I) is 

-dT~ —T ai{t)Hll > 

in which H' u = f\l/i*H'\l/idT, which can be integrated at once to 
give 

ai (t) = e~ 2 * iH ii t/h } ^t ^t', (39-7) 



XI-39a] TIME-DEPENDENT PERTURBATION 297 

the assumption being made that a* = 1 at the time t = 0. 

This expression shows the way that the coefficient ai changes 
during the time that the perturbation is acting. During this 
time the wave function, neglecting the terms with m ?± l } is 

a,Wl = ife-^'^' 
It will be observed that the time-dependent factor contains the 
first-order energy W° + H[ u as given by the Schrodinger per- 
turbation theory; this illustrates the intimate relation of the 
two perturbation theories. 

Now let us consider the remaining equations of the set 39-6, 
determining the behavior of the coefficients a m (t) with m 5* I. 
Replacing ai on the right side of 39-6 by its initial value 1 
ai(0) = 1, and neglecting all other a n 's, we obtain the set of 
approximate equations 

da m (t) 2wi 



- f ¥£*#'* 



dt ~ h ■-"'*'* 



This can be written as 

da m (t) = ~™H' ml e * dt, <^t ^t',m*l, 

in which 

H' ml = WH'tfdr, (39-8) 

and H' m i is independent of t, since we have considered W to be 
independent of / during the period <J / ^ V , and have replaced 
the time-containing wave functions Mf£* an d ^? by the amplitude 
functions \[/%* and \pi and the corresponding time factors, the 
latter being now represented explicitly by the exponential 
functions. These equations can be integrated at once; on intro- 
ducing the limits, and noting that a m (0) =0 for m t*1 % we 
obtain 

2iri(W m - Wi)t' 

o.(0 = H'J- ~^ m _ h Wi , m*l, (39-9) 

in which, it is remembered, the subscript I refers to the state 
initially occupied and m to other states. In case that the time 
V is small compared with the time h/(W m — Wi) y the expression 
can be expanded, giving 

1 The expression for ai(t) given by Equation 39-7 could be introduced in 
place of gu(0) ~ 1, with, however, no essential improvement in the result. 



298 PERTURBATION THEORY INVOLVING THE TIME [XI-39a 

ajfi « -X^^' m * l (3& " 10) 

At the time t' the wave function for the system (which was 
tyf at time t = 0) is approximately 

*(*') = ai (t')*f + %'a m (0*°n> (39-11) 

m 

(the prime on the summation sign indicating that the term 
m = I is omitted), in which aj is nearly equal to 1 and the a m 's 
are very small. This wave function continues to represent the 
system at times later than t', so long as it remains isolated. We 
could now carry out a measurement (of the energy, say) to deter- 
mine the stationary state of the system. The probability that 
the system would be found in the rath stationary state is a£a m . 

This statement requires the extension of the postulates regarding the 
physical interpretation of the wave equation given in Sections 10c and I2d. 
It was shown there that an average value could be predicted for a dynamical 
function for a system at time t from a knowledge of the wave function 
representing the system. The average value predicted for the energy of a 

system with wave function ¥ = 2^.a n *l is W = 2^cL* a nWl. However, an 

n n 

actual individual measurement of the energy must give one of the values 
Wq, W\, W% etc., inasmuch as it is only for wave functions corresponding 
to stationary states that the energy has a definite value (Sec. 10c). Hence 
when a measurement of the energy has been made, the wave function 
representing the system is no longer V = 2 an ^°> DU ^ ^ one °* tne functions 

*J, *J, *?, etc. 

This shows how a wave function does not really represent the system 
but rather our knowledge of the system. At time t — we knew the energy 
of the system to be TFJ, and hence we write ^t\ for the wave function. (We 
do not know everything about the system, however; thus we do not know 
the configuration of the system but only the probability distribution func- 
tion ¥j*¥{.) At time V we know that at time t = the wave function was 
¥j, and that the perturbation H' was acting between times t — and V. 
From this information we obtain the wave function of Equations 39-11, 
39-10, and 39-8 as representing our knowledge of the system. With it we 
predict that the probability that the system is in the mth stationary state 
is a*a m . So long as we leave the system isolated, this wave function repre- 
sents the system. If we allow the system to be affected by a known pertur- 
bation, we can find a new wave function by the foregoing methods. If we 
now further perturb the system by an unknown amount in the process of 
making a measurement of the energy, we can no longer apply these methods; 



XI-40a] EMISSION AND ABSORPTION OF RADIATION 299 

instead, we assign to the system a new wave function compatible with our 
new knowledge of the result of the experiment. 

A more detailed discussion of these points will be given in Chapter XV. 

Equation 39-10 shows that in case t' is small the probability of 
finding the system in the stationary state m as a result of transi- 
tion from the original state I is 

a:a m = ^H'^H' ml t", (39-12) 

being thus proportional to the square of the time t r rather than 
to the first power as might have been expected. In most cases 
the nature of the system is such that experiments can be designed 
to measure not the probability of transition to a single state but 
rather the integrated probability of transition to a group of 
adjacent states; it is found on carrying out the solution of the 
fundamental equations 39-6 and subsequent integration that for 
small values of t' the integrated probability of transition is pro- 
portional to the first power of the time t'. An example of a 
calculation of a related type will be given in Section 406. 

40. THE EMISSION AND ABSORPTION OF RADIATION 

Inasmuch as a thoroughly satisfactory quantum-mechanical 
theory of systems containing radiation as well as matter has not 
yet been developed, we must base our discussion of the emission 
and absorption of radiation by atoms and molecules on an 
approximate method of treatment, drawing upon classical electro- 
magnetic theory for aid. The most satisfactory treatment of 
this type is that of Dirac, 1 which leads directly to the formulas 
for spontaneous emission as well as absorption and induced 
emission of radiation. Because of the complexity of this theory, 
however, we shall give a simpler one, in which only absorption 
and induced emission are treated, prefacing this by a general 
discussion of the Einstein coefficients of emission and absorption 
of radiation in order to show the relation that spontaneous 
emission bears to the other two phenomena. 

40a. The Einstein Transition Probabilities. — According to 
classical electromagnetic theory, a system of accelerated electri- 
cally charged particles emits radiant energy. In a bath of 

*P. A. M. Dirac, Proc. Roy. Soc. A112, 661 (1926); A114, 243 (1927); 
J. C. Slater, Proc. Nat. Acad. Set. 13, 7 (1927). 



300 PERTURBATION THEORY INVOLVING THE TIME [XI-40a 

radiation at temperature T it also absorbs radiant energy, the 
Tates of absorption and of emission being given by the classical 
laws. These opposing processes might be expected to lead to a 
state of equilibrium. The following treatment of the correspond- 
ing problem for quantized systems (atoms or molecules) was 
given by Einstein 1 in 1916. 

Let us consider two non-degenerate stationary states m and n 
of a system, with energy values W m and W n such that W m is 
greater than W n . According to the Bohr frequency rule, transi- 
tion from one state to another will be accompanied by the 
emission or absorption of radiation of frequency 

_ W m - Wn 
Vmn ~ h 

We assume that the system is in the lower state n in a bath of 
radiation of density p(y mn ) in this frequency region (the energy 
of radiation between frequencies v and v + dv in unit volume 
being p{v)dv). The probability that it will absorb a quantum 
of energy of radiation and undergo transition to the upper state 
in unit time is 

B n -+mp(Vmn)- 

B n -* m is called Einstein's coefficient of absorption. The proba- 
bility of absorption of radiation is thus assumed to be propor- 
tional to the density of radiation. On the other hand, it is 
necessary in order to carry through the following argument to 
postulate 2 that the probability of emission is the sum of two 
parts, one of which is independent of the radiation density and 
the other proportional to it. We therefore assume that the 
probability that the system in the upper state m will undergo 
transition to the lower state with the emission of radiant energy 
is 

A m ~*n ~T ■t>m~>nP\Vmn)» 

A m ->n is Einstein 7 s coefficient of spontaneous emission and 2? m -> n 
is Einstein 9 s coefficient of induced emission. 

1 A. Einstein, Verh. d. Deutsch. Phijs. Ges. 18, 318 (1916); Phys. Z. 18, 
121 (1917). 

2 This postulate is of course closely analogous to the classical theory, 
according to which an oscillator interacting with an electromagnetic wave 
could either absorb energy from the field or lose energy to it, depending 
on the relative phases of oscillator and wave. 



XI-40a] EMISSION AND ABSORPTION OF RADIATION 301 

We now consider a large number of identical systems of this 
type in equilibrium with radiation at temperature T. The 
density of radiant energy is known to be given by Planck's 
radiation law as 

p(v) =-- r -- 5 - , (40-1) 

e kf - 1 

in which k is the Boltzmann constant. Let the number of sys- 
tems in state m be N m , and that in state n be N n . The number of 
systems undergoing transition in unit time from state n to state 
m is then 

N n Bn-+mp(v mn ) y 

and the number undergoing the reverse transition is 

N m {A m -+n + B m -> n p(v mn )}. 

At equilibrium these two numbers are equal, giving 

iVffi & n -+mP\ymn) 

The equations of quantum statistical mechanics (Sec. 49) 
require that the ratio N n /N m be given by the equation 

ip = e kT = e h "»» /kT . (40-3) 

From Equations 40-2 and 40-3 we find for p(v mn ) the expression 

A _> 

p( v ^) = £ ^J£"_. B ( 40 ~4) 



'm—m 



In order for this to be identical with Equation 40-1, we must 
assume that the three Einstein coefficients are related by the 
equations 

B n ->m = B m -+n (40-"6o) 

and 

A m ^ n = ^^B m ->n; (40-56) 



that is, the coefficients of absorption and induced emission are 
equal and the coefficient of spontaneous emission 1 differs from 
them by the factor fhchv^Jc*. 

1 It is interesting to note that at the temperature T = •— — - the proba- 

k log 2 

bilities of spontaneous emission and induced emission are equal. 



302 PERTURBATION THEORY INVOLVING THE TIME [XI-40b 

40b. The Calculation of the Einstein Transition Probabilities 
by Perturbation Theory. — According to classical electromagnetic 
theory, the density of energy of radiation of frequency v in space, 
with unit dielectric constant and magnetic permeability, is 
given by the expression 

P« = ^TO, (40-*) 

in which E\v) represents the average value of the square of the 
electric field strength corresponding to this radiation. The 
distribution of radiation being isotropic, we can write 

%W$) = ^ = TO = Wy), (40-7) 

E x {y) representing the component of the electric field in the 
x direction, etc. We may conveniently introduce the time 
variation of the radiation by writing 

E x (v) = 2E° x (v) cos 2wvt = E° x (v)(e** M + e-*™'), (40-8) 

the complex exponential form being particularly convenient for 
calculation. Since the average value of cos 2 2wvt is J^, we see 
that 

pW = ±W(v) = I'm*) = £tf."«. (40-9) 

Let us now consider two stationary states m and n of an 
unperturbed system, represented by the wave functions ^ and 
^£, and such that W m is greater than W n . Let us assume that 
at the time t = the system is in the state n, and that at this 
time the system comes under the perturbing influence of radiation 
of a range of frequencies in the neighborhood of v mn , the electric 
field strength for each frequency being given by Equation 40-8. 
We shall calculate the probability of transition to the state m 
as a result of this perturbation, using the method of Section 39. 
The perturbation energy for a system of electrically charged 
particles in an electric field E x parallel to the x axis is 

W = E x ^x u (40-10) 

in which e,- represents the charge and x,- the x coordinate of the 
jth particle of the system. The expression ]xe;£, (the sum being 



XI-40b] EMISSION AND ABSORPTION OF RADIATION 303 

taken over all particles in the system) is called the component of 
electric dipole moment of the system along the x axis and is often 
represented by the symbol /**. We now make the approximation 
that the dimensions of the entire system (a molecule, say) are 
small compared with the wave length of the radiation, so that 
the electric field of the radiation may be considered constant over 
the system. In the case under consideration the field strength E x 
is given by the expression 

E x = $El{v)(e 2 * ivt + e- 2 ™ l )dv. 

Let us temporarily consider the perturbation as due to a single 
frequency v. Introducing a m (0) = and a n (0) = 1 in the right 
side of Equation 39-6 (a n being the coefficient of a particular 
state and all the other coefficients in the sum being zero), this 
equation becomes 

MO = ~f*l*H'*°dr - -~ji:*e^ W ^ x ( v )(e^> + 

e -2*ivt} J^ qxtfle h " dr. 



If we now introduce the symbol n Zmn to represent the integral 

M* W n = fK*Xe 3 xMdr = fr m W»dr, (40-11) 

we obtain the equation 



da m (t) 
dt 


9 i ( 2ir1 


'(Wm-Wn 


> + kv)t 


+ 


2xi 

e h 


(W n - 


which gives, 


on in 


tegration, 
'1 .T^- 


■Wn+K*)t 










a m (t) = n Xmn El(v)\ 


1 1 — e n 


+ hv 


+ 














1 - 


e h 


-Wn 


-hp)i* 



W»-hp)t\ 

r 



W m ~ Wn 



(40-12) 



Of the two terms of Equation 40-12, only one is important, 
and that one only if the frequency v happens to lie close to 
Vmn = (W m — W n )/h. The numerator in each fraction can vary 
in absolute magnitude only between and 2, and, inasmuch as 
for a single frequency the term n Xmn El(v) is always small, the 



304 PERTURBATION THEORY INVOLVING THE TIME [XI-40b 

expression will be small unless the denominator is also very 
small; that is, unless hv is approximately equal to W m — W n . 
In other words, the presence of the so-called resonance denomina- 
tor Wm — Wn — hv causes the influence of the perturbation 
in changing the system from the state n to the higher state m 
to be large only when the frequency of the light is close to that 
given by the Bohr frequency rule. In this case of absorption, it 
is the second term which is important; for induced emission of 
radiation (with W m — W n negative), the first term would play 
the same role. 

Neglecting the first term, we obtain for a*(t)a m (t), after slight 
rearrangement, the expression 



SUlt&Wm - Wn- kv)t\ 

a*(t)a m {t) = 4( M J% 02 (,)- ( j 



(W m ~Wn- hvY 

(If ji Xmn is complex, the square of its absolute value is to be used 
in this equation.) This expression, however, includes only the 
terms due to a single frequency. In practice we deal always 
with a range of frequencies. It is found, on carrying through 
the treatment, that the effects of light of different frequencies are 
additive, so that we now need only to integrate the above 
expression over the range of frequencies concerned. The 
integrand is seen to make a significant contribution only over the 
region of v near v mny so that we are justified in replacing El(v) 
by the constant El(v mn ), obtaining 



a*(t)a m (t) = 4U ! ^ ! ('..) 



/sin 2 t(W m -W n - hv)t\ 
(W m -W n - hvY ° 



This integral can be taken from — °o to + °° , inasmuch as the 
value- of the integrand is very small except in one region; and 

I sin^ x 
making use of the relation I — ^— dx = t, we can obtain the 

J— » # 

equation 

a*(t)a m (t) = ^0O^2 f (O*- (40-13) 

It is seen that, as the result of the integration over a range of 
values of v } the probability of transition to the state m in time t 
is proportional to t f the coefficient being the transition probability 



S3-40b] EMISSION AND ABSORPTION OF RADIATION 305 

as usually defined. With the use of Equation 40-9 we may now 
introduce the energy density p(v mn ), obtaining as the probability 
of transition in unit time from state n to state m under the 
influence of radiation polarized in the x direction the expression 

3^2(Mx m n) 2 p(^mn). 

The expressions for the y and z directions are similar. Thus we 
obtain for the Einstein coefficient of absorption B n -> m the 
equation 

B n ^ m = |g{ ( M ,„„) 2 + GO' + W ! | (40-14a) 

By a completely analogous treatment in which the values 
«n(0) = 0, a m (0) = 1 are used, the Einstein coefficient of induced 
emission B m -> n is found to be given by the equation 

B m ^ n = |p{(Mx.J 2 + U.J 2 + U.n) 2 }, (40-146) 

as, indeed, is required by Equation 40-5a. 

Our treatment does not include the phenomenon of spontaneous 
emission of radiation. Its extension to include this is not easy; 
Dirac's treatment is reasonably satisfactory, and we may hope 
that the efforts of theoretical physicists will soon provide us with 
a thoroughly satisfactory discussion of radiation. For the 
present we content ourselves with using Equation 40-56 in 
combination with the above equations to obtain 

A m ^ n = i|jgk{0O« + (*..)* + GO*} (40-15) 

as the equation for the Einstein coefficient of spontaneous 
emission. 

As a result of the foregoing considerations, the wave-mechanical 
calculation of the intensities of spectral lines and the determina- 
tion of selection rules are reduced to the consideration of the 
electric-moment integrals defined in Equation 40-11. We shall 
discuss the results for special problems in the following sections. 

It is interesting to compare Equation 40-15 with the classical 
expression given by Equation 3-4 of Chapter I. Recalling that 
the energy change associated with each transition is kv mn , we 
see that the wave-mechanical expression is to be correlated with 



306 PERTURBATION THEORY INVOLVING THE TIME [XI-40d 

the classical expression for the special case of the harmonic 
oscillator by interpreting Mx™ as one-half the maximum value 
exo of the electric moment of the classical oscillator. 

40c. Selection Rules and Intensities for the Harmonic Oscil- 
lator. — The electric dipole moment for a particle with electric 
charge e carrying out harmonic oscillational motion along the 
x axis (a neutralizing charge —e being at the origin) has the 
components ex along this axis and zero along the y and z axes. 
The only non-vanishing dipole moment integrals /**•« = &c m n 
have been shown in Section lie to be those with m = n + 1 or 
m = n — 1. Hence the only transitions which this system can 
undergo with the emission or absorption of radiation are those 
from a given stationary state to the two adjacent states. 1 The 
selection rule for the harmonic oscillator is therefore An = ±1, 
and the only frequency of light emitted or absorbed is v . The 
expression for z n ,n-i in Equation ll-25a corresponds to the 
value 

with a — 4v 2 mvo/h y for the coefficient of spontaneous emission, 
with similar expressions for the other coefficients. An applica- 
tion of this formula will be given in Section 40e. 

Problem 40-1. Show that for large values of n Equation 40-16 reduces 
to the classical expression for the same energy. 

Problem 40-2. Discuss the selection rules and intensities for the three 
dimensional harmonic oscillator with characteristic frequencies v Xi vy 9 
and v t . 

Problem 40-3. Using first-order perturbation theory, find perturbed 
wave functions for the anharmonic oscillator with V = 2ic i mvlx 2 -+■ ax 3 , 
and with them discuss selection rules and transition probabilities. 

40d. Selection Rules and Intensities for Surface -harmonic 
Wave Functions. — In Section 18 we showed that the wave 
functions for a system of two particles interacting with one 
another in the way corresponding to the potential function 

1 This statement is true only to within the degree of approximation of our 
treatment. A more complete discussion shows that transitions may also 
occur as a result of interactions corresponding to quadrupole terms and still 
higher terms, as mentioned in Section 3, and as a result of magnetic 
interactions. 



XI-40d] EMISSION AND ABSORPTION OF RADIATION 307 

V(r), in which r is the distance between the two particles, are 
of the form 

in which the &,<p functions Bim(&)$m(<p) are surface harmonics, 
independent of V(r). We can hence discuss selection rules and 
intensities in their dependence on I and m for all systems of 
this type at one time. 

The components of electric dipole moment along the x, y, 
and z axes are 

Mx = n(r) sin # cos <p, 

Mi/ = n(r) sin # sin <p, 
and 

\x t = /x(r) cos #, 

in which n(r) is a function of r alone, being equal to er for two 
particles with charges +e and — e. Each of the dipole moment 
integrals, such as 

"..„.„.„„.. = ///BSWe,* (*)«(^) M (r) sin # cos *> 
i2*'j<(r)6rm' (#)<l> m ^)r 2 sin #d<pd#dr, 

can accordingly be written as the product of three factors, one 
involving the integral in r, one the integral in #, and one the 
integral in <p: 

»V n i^l> m , = M»i»'i/» tal , w ,fo_,,> (40-17) 

in which 

Hnin>i> = J[ °°^(r) M (r)i? n ,Kr)r 2 dr, (40-18) 

/v — 7 = I e '-W 8in *>Qi'*'W sin &», (40-19) 

'*— ') ^2 ( COS *>) 

*-'[ = X T ^ ( ^) sin ^*«'W^- ( 4 °- 2 °) 

(In Equations 40-19 and 40-20 the subscripts x, y, and z are 
respectively associated with the three factors in braces.) 

Let us first discuss the light polarized along the z axis, cor- 
responding to the dipole moment n t . From the orthogonality 



and 



308 PERTURBATION THEORY INVOLVING THE TIME [XI-40d 

and normalization integrals for $(<p) we see that 
Qz = for m! 7* m 

° mm' 

and 

Qtmm = 1. 
In discussing /* we consequently need consider only the 
integrals with m' = m. It is found with the use of the recursion 
formula (Prob. 19-2) 

COS tfP[">(COS 0) = ^+Mp lml i(cog 0) + 

- ~{2lt 1) 1) p '"' l(cOS *> (40_21) 
that fz lmVm , vanishes except when V is equal to I + 1 or I — 1. 

A similar treatment of the integrals for x and 2/ shows that 
light polarized along these axes is emitted only when m changes 
by +1 or — 1, and I changes by +1 or — 1. 

We have thus obtained the selection rules Am = 0, +1, or —1 
and AZ = +1 or — 1. The selection rule for I is discussed in 
the following sections. That for m can be verified experimentally 
only by removing the degeneracy, as by the application of a 
magnetic field; it is found, in agreement with the theory, that 
in the Zeeman effect the light corresponding to Am = is 
polarized along the z axis (the axis of the magnetic field), and that 
corresponding to Am = ± 1 is polarized in the xy plane. 

The values of the products of the factors / and g are 

(M - i(fiis - i/(* + MX* + M - D l* \ 

WJ W».ii-» - %ug)v - 2\ (21 + l)(2l - 1) j \ 

m - i(fQ) l/ (^ H)g - H + 1) H (40-22) 

_ / (i + M)(Mm|) y* 

utf; ^i-i^M«i - \ (a + l)(2l - 1) / 
with similar expressions for the transitions I to I + 1, etc. 

Problem 40-4. Using Equation 40-21, obtain selection rules and 
intensities for ju*. 

Problem 40-6. Similarly derive the other formulas of 40-22. 

Problem 40-6. Calculate the total probability of transition from one 
level with given value of / to another, by summing over m. By separate 
summation for /x x , n Vf and m* show that the intensity of light polarized along 
these axes is the same. 



XI-40e] EMISSION AND ABSORPTION OF RADIATION 309 

40e. Selection Rules and Intensities for the Diatomic Mole- 
cule. The Franck -Condon Principle. — A very simple treatment 
of the emission and absorption of radiation for the diatomic 
molecule can be given, based on the approximate wave functions 
of Section 35c. For the complex system of two nuclei and several 
electrons the electric dipole moment n(r) can be expanded as a 
series in r — r , 

M(r) = mo + €(r - r ) + • • • , (40-23) 

in which e is a constant. The permanent dipole moment /xo is 
the quantity which enters in the theory of the dielectric constant 
of dipole molecules; its value is known from dielectric constant 
measurements for many substances. 

Introducing this expansion in Equation 40-18, we find as a 
first approximation that n may change by or by ±1. In the 
former case the emission or absorption of radiation is due to 
the constant term fi , and in the latter case to the term e(r — r ), 
the integrals being then similar to the harmonic oscillator 
integrals. The values of ju nn / are 

linn = mo (40-24a) 

and 

Mn.n-1 - t^f (40-246) 

in which a = ^T 2 fxv /h (/x being here the reduced mass for the 
molecule). The selection rules and intensity factors for I and 
m are as given in the preceding section. 

It is found experimentally that dipole molecules such as the 
hydrogen halides absorb and emit pure rotation and oscillation- 
rotation bands in accordance with these equations. In all 
these bands the selection rule Al = ± 1 is obeyed, and Zeeman- 
effect measurements have shown similar agreement with the 
selection rule for m. The intensities of lines in the pure rotation 
bands show rough quantitative agreement with Equation 
40-24a, using the dielectric constant value of /* , although because 
of experimental difficulties in the far infrared the data are as 
yet not very reliable. Measurements of absorption intensities 
for An = 1 have been used to calculate c As seen from the 
following table, « is of the order of magnitude of mo/Vo, so that 
these molecules may be considered roughly as equivalent to two 
particles of charges +€ and — €. 



310 PERTURBATION THEORY INVOLVING THE TIME [XI -40© 



Table 40-1 



Mo (dielectric constant) 



r 



»o/r 



HC1 
HBr 
HI 



1.034 X 10- 18 e.s.u. 

0.788 

0.382 



1.28 A 

1.42 

1.62 



0.169 e 
.116 
.049 



0.086 e 
.075 
.033 



i E. BabtholomjS, Z. phya. Chem. B 23, 131 (1933). 

It is also observed that oscillation-rotation bands with 
An = 2, 3, etc. occur; this is to be correlated with the deviation 
of the potential function V(r) from a simple quadratic function. 

In the foregoing discussion we have assumed the electronic 
state of the molecule to be unchanged by the transitions. The 
selection rule for n and the intensities are different in case there is a 
change in the electronic state, being then determined, according 
to the Franck-Condon principle, 1 mainly by the nature of the 
electronic potential functions for the two electronic states. As 
we have seen in Section 34, there is little interaction between the 
electronic motion and the nuclear motion in a molecule, and 
during an electronic transition the internuclear distance and 
nuclear velocities will not change very much. Let us consider 
the two electronic states A and B, represented by the potential 
curves of Figure 40-1, in which the oscillational levels are also 
shown. If the molecule is in the lowest oscillational level 
n' = of the upper state, the probability distribution function 
for r is large only for r close to r v We would then expect an 
electronic transition to state B to leave the molecule at about the 
point Pi on -the potential curve, the nuclei having only small 
kinetic energy; these conditions correspond to the levels n" = 7 
or 8 for the lower state. 

This simple argument is justified by wave-mechanical con- 
siderations. Let us consider that the wave functions for the 
upper electronic state may be written as &>$»', in which fa* 
represents the nuclear oscillational part of the wave function 
described by the quantum number n', and ty a > the rest of the wave 
function (electronic and nuclear rotational), the symbol or' 
representing all other quantum numbers. Similarly, we write 
^r"^n" for the wave functions for the lower electronic state. 

1 J. Fbanck, Trans. Faraday Soc, 21, 536 (1926); E. U. Condon, Phys- 
Rev. 28, 1182 (1926); 32, 868 (1928). 



XI-40e] EMISSION AND ABSORPTION OF RADIATION 311 

The electric dipole moment integrals /**,,* v»"> *V*w> an( * 
H s , nV , M „ are of the form 

^Vnw = /^VnVx^-^n-dr. (40-25) 

We assume that in this case, when there is a change in the 



w 




r eA 



r AB 
Fig. 40-1. — Energy curves for two electronic states of a molecule, to illustrate 
the Franck-Condon principle. 

electronic state, the dipole moment function /* changes only 
slowly with change in the internuclear separation r, being deter- 
mined essentially by the electronic coordinates. Neglecting the 



312 PERTURBATION THEORY INVOLVING THE TIME [XI-40f 

dependence of /z on r, we can then integrate over all coordinates 
except r, obtaining 

*V.w = M v JW»-rWr. (40-26) 

The integral in r, determining the relative intensities of the 
various n'n" bands, is seen to have the form of an orthogonality 
integral in r. Hence if the two potential functions Va and V B 
were identical except for an additive constant the integral would 
vanish except for n' = n", the selection rule for n then being 
An = o. In the case represented by Figure 40-1, the wave func- 
tion i/v with n' = is large only in the neighborhood of r = r ejL . 
The wave functions i> n >> with n" = 7 or 8 have large values in 
this region, so that the bands n f = — > n" — 7 or 8 will be 
strong. The intensity of the bands for smaller or larger values 
of n" will fall off. For smaller values of n" the functions i/v 
show the rapid exponential decrease in the region near r e ^ 
(corresponding to the fact that the classical motion of the nuclei 
would not extend into this region); whereas for larger values 
of n" the functions \p n >> show a rapid oscillation between positive 
and negative values, causing the integral with the positive 
function yf/ n > with n r — to be small (the oscillation of ^v 
between positive and negative values corresponding to large 
nuclear velocities in the classical motion). 

Similarly the transitions from the level with n' = 5, the wave 
function for which has its maximum values near the points 
P 2 and P 3 , will occur mainly to the levels n" = 2 or 3 and 
n" = 11 or 12. 1 

40f . Selection Rules and Intensities for the Hydrogen Atom. — 
The selection rule for Z, discussed in Section 40d, allows only 
transitions with AZ = ± 1 for the hydrogen atom. The lines of 
the Lyman series, with lower state that with n — 1 and I = 0, 
are in consequence due to transitions from upper states with 
1 = 1. The radial electric dipole moment integral 

Hn> n >> = $R* v {r)rR n >>i"{r)rHr 

has been evaluated by Pauli 2 for several special cases. For 

1 For a more complete discussion of this subject the reader is referred to 
the papers of Condon and to the discussions in Condon and Morse, " Quan- 
tum Mechanics/' Chap. V, and Ruark and Urey, "Atoms, Molecules and 
Quanta," Chap. XII. 

2 Communicated in Schrddinger's third 1926 paper. 



XI-40g] EMISSION AND ABSORPTION OF RADIATION 313 

n" = 1, V = its value corresponds to the total intensity, 
aside from a constant factor, of 

in'l = 



n'(n' + 1) 2 » +1 



This has a non- vanishing value for all values of n 1 greater than 1. 
Hence there is no selection rule for n for the Lyman series, all 
transitions being allowed. It is similarly found that there is no 
selection rule for n for spectral series in general. 

For the Balmer series, with lower state that with n = 2 and 
I = or 1, the selection rule for I permits the transitions — > 1, 
1 — >0, and 2 — > 1. The total intensity corresponding to these 
transitions from the level n = n f to n = 2 is, except for a constant 
factor, 

7 " - Mrr^ i3n ' 2 ~ 4)(5n ' 2 " 4) - 

The operation of the selection rule for I for hydrogen and 
hydrogenlike ions can be seen by the study of the fine structure 
of the lines. The phenomena are complicated, however, by the 
influence of electron spin. 1 In alkali atoms the levels with given 
n and varying I are widely separated, and the selection rule for I 
plays an important part in determining the nature of their 
spectra. Theoretical calculations have also been made of the 
intensities of lines in these spectra with the use of wave functions 
such as those described in Chapter IX, leading to results in 
approximate agreement with experiment. 

40g. Even and Odd Electronic States and Their Selection 
Rules. — The wave functions for an atom can all be classified as 
either even or odd. An even wave function of N electrons is 
one such that \p(xi, y u Zi, x^ • • • , Zn) is equal to \p(— Xi, —y u 
— Zi,—X2,'*', — s>r); that is, the wave function is unchanged 
on changing the signs of all of the positional coordinates of the 
electrons. An odd wave function is one such that t(zi, yi, 
ti,X2, - - • , zn) is equal to -^(-Xi, -y lf -z h —x 2 , • • • , —Zn). 

Now we can show that the only transitions accompanied by the 
emission or absorption of dipole radiation which can occur are 
those between an even and an odd state (an even state being one 
represented by an even wave function, etc.). The electric 

1 See Pauling and Goudsmit, "The Structure of Line Spectra," Sec. 16. 



314 PERTURBATION THEORY INVOLVING THE TIME [XI-41 

N 

moment component functions £\ex if etc. change sign in case 

that the electronic coordinates are replaced by their negatives. 
Consequently an electric-moment integral such as ^t'^ex^n^dr 

i 

will vanish in case that both ^v and ^ n " are either even or odd, 
but it is not required to vanish in case that one is even and the 
other odd. We thus have derived the very important selection 
rule that transitions with the emission or absorption of dipole 
radiation are allowed only between even and odd states. Because of 
the practical importance of this selection rule, it is customary to 
distinguish between even and odd states in the term symbol, by 
adding a superscript ° for odd states. Thus various even states 
are written as X >S, 3 P, 2 D, etc., and odd states as 1 5°, 3 P°, 2 D°, etc. 

In case that the electronic configuration underlying a state is 
known, the state can be recognized as even or odd. The one- 
electron wave functions are even for I = 0, 2, 4, etc. (s, d, g, etc., 
orbitals) and odd for I = 1, 3, 5, etc. (p, /, h y etc., orbitals). 
Hence the configuration leads to odd states if it contains an 
odd number of electrons in orbitals with I odd, and otherwise to 
even states. For example, the configuration ls 2 2s 2 2p 2 leads to the 
even states 1 S ) l D, and 3 P, and the configuration ls 2 2p3d to 
the odd states »P°, l D°, l F° y 3 P°, 3 D°, and 3 P°. 

Even and odd states also occur for molecules, and the selection 
rule is also valid here. A further discussion of this point will be 
given in Section 48. 

Problem 40-7. Show that the selection rules forbid a hydrogen atom 
in a rectangular box to radiate its translational kinetic energy. Extend the 
proof to any atom in any kind of box. 

41. THE RESONANCE PHENOMENON 

The concept of resonance played an important part in the dis- 
cussion of the behavior of certain systems by the methods of 
classical mechanics. Very shortly after the discovery of the 
new quantum mechanics it was noticed by Heisenberg that a 
quantum-mechanical treatment analogous to the classical 
treatment of resonating system can be applied to many problems, 
and that the results of the quantum-mechanical discussion in these 
cases can be given a simple interpretation as corresponding to a 
quantum-mechanical resonance phenomenon. It is not required 



XI -41a] 



THE RESONANCE PHENOMENON 



315 



that this interpretation be made; it has been found, however, 
that it is a very valuable aid to the student in the development 
of a reliable and productive intuitive understanding of the 
equations of quantum mechanics and the results of their applica- 
tion. In the following sections we shall discuss first classical 
resonance and then resonance in quantum mechanics. 

41a. Resonance in Classical Mechanics. — A striking phe- 
nomenon is shown by a classical mechanical system consisting of 
two parts between which there is operative a small interaction, 
the two parts being capable of executing harmonic oscillations 
with the same or nearly the same frequency. It is observed 
that the total oscillational energy fluctuates back and forth 




Fig. 41-1.- 



-A system of coupled pendulums, illustrating the phenomenon of 
resonance. 



between the two parts, one of which at a given time may be 
oscillating with large amplitude, and at a later time with small 
amplitude, while the second part has changed jn the opposite 
direction. It is customary to say that the two parts of the 
system are resonating. A familiar example of such a system is 
composed of two similar tuning forks attached to the same base. 
After one fork is struck, it gradually ceases to oscillate, while at 
the same time the other begins its oscillation. Another example 
is two similar pendulums connected by a weak spring, or attached 
to a common support in such a way that interaction of the two 
occurs by way of the support (Fig. 41-1). It is observed that if 
only one pendulum is set to oscillating, it will gradually die down 
and stop, while the other begins to oscillate, ultimately reaching 
the amplitude of oscillation initially given the first (neglecting the 



(41-2) 



316 PERTURBATION THEORY INVOLVING THE TIME [XI-41a 

f rictional dissipation of energy) ; and that this process of transfer 
of energy from one pendulum to the other is repeated over and 
over. 

It is illuminating to consider this system in greater detail. 
Let Zi and x 2 be the coordinates for two oscillating particles each 
of mass m (such as the bobs of two pendulums restricted to 
small amplitudes, in order that their motion be harmonic), 
and let v be their oscillational frequency. We assume for the 
total potential energy of the system the expression 

V(xi, x 2 ) = 2-K <l mv\x\ + 2-w 1 mv\x\ + 4tT 2 m\x 1 x 2j (41-1) 

in which 4tTr 2 m\xiX2 represents the interaction of the two oscil- 
lators. This simple form corresponds to a Hooked-law type of 
interaction. The solution of the equations of motion is easily 
accomplished by introducing the new variables 1 

£ = T/^ Xl +*2),J 

In terms of these, the potential energy becomes 

F(£, v ) = Wm{v\ + \)e + 2*hn{v\ - X)t? 2 , 

while the kinetic energy has the form 

T = V 2 mx\ + }/ 2 mxl = y 2 m¥ + y 2 mi)\ 

These expressions correspond to pure harmonic oscillation of the 
two variables £ and r\ (Sec. la), each oscillating with constant 
amplitude, £ with the frequency \/v\ + X and r) with the 
frequency y/ v% — X, according to the equations 

£ = £0 cos {2iry/vl + X t + bz)\ (41-3) 

7] = Tjo cos (2ttV^ - X t + 5,). ) V 

From these equations we obtain the equations 

Xi = -^ cos {2W»l + X t) + -5?= cos (27rV^T =r X"0,) 

Y 3 V2 (41-4) 

x 2 = ~ cos (fcrVVj + X t) - -52= cos (2ttv^§ - X 0,1 

1 These are the normal coordinates of the system, discussed in Section 37. 



XI-41a] THE RESONANCE PHENOMENON 317 

for xi and x 2 , in which we have put the phase constants 8$ and 8, 
equal to zero, as this does not involve any loss in generality- 
It is seen that for \/vl small the two cosine functions differ only 
slightly from one another and both x\ and x 2 carry out approxi- 
mate harmonic oscillation with the approximate frequency vq, 
but with amplitudes which change slowly with the time. Thus 
at t ^ the cosine terms are in phase, so that x\ oscillates with 
the amplitude (£ + *7o)/\/2 and X2 with the smaller amplitude 
(£o — i/o)/v / 2- At the later time t = h such that 

the cosine terms are just out of phase, Xi then oscillating with the 

amplitude (£ — rj )/\/2 and x 2 with the amplitude (£ + ^7o)/v / 2- 
Thus we see that the period r of the resonance, the time required 
for xi to change from its maximum to its minimum amplitude 
and then back to the maximum, is given by the equation 



V^f+Xr = a/^T-Xt + 1 
or 

1 ~ "o 



(41~5) 



It is also seen that the magnitude of the resonance depends on 
the constants of integration £o and rj , the amplitudes of motion 
of Xi and x 2 varying between the limits \/2£o and in case that 
Vo = £o, and retaining the constant value £o/\/2 (no resonance!) 
in case that r) = 0. 

The behavior of the variables x± and x 2 may perhaps be 
followed more clearly by expanding the radicals y/v\ + X and 
V^o — X in powers of \/v% and neglecting terms beyond the 
first power. After simple transformations, the expressions 
obtained are 

Xl = (&±Jo) cos27r-^ cos 27rvot - iksiJ!S> s in 2tt-* sin 2™rf 
V2 "o y/2 v o 

and 

X2 = ^°~ 71o) cos27r-tcos2irv t - ( *° + no) sin 2tt-* sin 2™<>*. 
V2 ^o V2 "o 

It is clear from this treatment that we speak of resonance only 
because it is convenient for us to retain the coordinates Xi and 



318 PERTURBATION THEORY INVOLVING THE TIME (XI-41b 

Xi in the description of the system; that is, to speak of the motion 
of the pendulums individually rather than of the system as a 
whole. We can conceive of an arrangement of levers whereby 
an indicator in an adjacent room would register values of £, and 
another values of 77. An observer in this room would say that 
the system was composed of two independent harmonic oscillators 
with different frequencies and constant amplitudes, and would 
not mention resonance at all. 

Despite the fact that we are not required to introduce it, the 
concept of resonance in classical mechanical systems has been 
found to be very useful in the description of the motion of sys- 
tems which are for some reason or other conveniently described 
as containing interacting harmonic oscillators. It is found 
that a similar state of affairs exists in quantum mechanics. 
Quantum-mechanical systems which are conveniently considered 
to show resonance occur much more often, however, than 
resonating classical systems, and the resonance phenomenon 
has come to play an especially important part in the applications 
of quantum mechanics to chemistry. 

41b. Resonance in Quantum Mechanics. — In order to illus- 
trate the resonance phenomenon in quantum mechanics, let us 
continue to discuss the system of interacting harmonic oscil- 
lators. 1 Using the potential function of Equation 41-1, the wave 
equation can be at once separated in the coordinates £ and 17 
and solved in terms of the Hermite functions. The energy 
levels are given by the expression 

W n ^ = (n* + M)hV~vJ+\ + (n, + M)hVpf=\, (41-6) 

which for X small reduces to the. approximate expression 

F,^ S (n + 1)A„ + (n« - O^ - ^^! + . . . , 

(41-7) 

in which n = n^ + fly,. The energy levels are shown in Figure 
41-2; for a given value of n there are n + 1 approximately 
equally spaced levels. 

This treatment, like the classical treatment using the coordi- 
nates £ and r), makes no direct reference to resonance. Let us 

1 This example was used by Heisenberg in his first papers on the resonance 
phenomenon, Z.f. Phys. 38, 411 (1926); 41. 239 (1927). 



XI -41b] 



THE RESONANCE PHENOMENON 



319 



now apply a treatment in which the concept of resonance enters, 
retaining the coordinates Xi and x 2 because of their familiar 
physical interpretation and applying the methods of approximate 
solution of the wave equation given in Chapters VI and VII; 
indeed, if the term in X were of more complicated form, it would 
be necessary to resort to some approximate treatment. This 
term is conveniently considered as the perturbation function in 
applying the first-order perturbation theory. The unperturbed 



Ahv c 



W 



2hi>< 



hv. 



n-3 



i 



<■ 



n r n n 

- 1 
--1 

--3 

- 2 

- 
— 2 



< 



Fig. 41-2. — Energy levels for coupled harmonic oscillators; left, with X » 0; 
right, with X « vl/b, 

wave equation has as solutions products of Hermite functions 
in xi and x 2 , 

<n,(*l, X*) = r ni (xM 2 (*2) = 

2 t 

— fifL 1 _ aXi 

N ni H ni (V*ci)e 2 N nt H ni (\^x 2 )e 2 , (41-8) 
corresponding to the energy values 

WTU = (ni + n 2 + l)hp Q = (n + l)hv 0f 

with n = tti + n 2 , (41-9) 

the nth level being (n + l)-fold degenerate. 

The perturbation energy for the non-degenerate level n = is 
zero. For the level n = 1 the secular equation is found to be 
(Sec. 24) 



320 PERTURBATION THEORY INVOLVING THE TIME [XI-41b 

h\ 



£- -W 



= 0, 



giving W = ±ftX/2^ . A similar treatment of the succeeding 
degenerate levels shows that the first-order perturbation theory 
leads to values for the energy expressed by the first two terms 
of Equation 41-7. 

The correct zeroth-order wave functions for the two levels 
with n = 1 are found to be 



and 



V2 



* A = ^ mxM{x2) ~ ^(ziWfe)}, 



xps corresponding to the lower of the two levels and \f/ A to the 
upper. The subscripts S and A are used to indicate that the 
functions are respectively symmetric and antisymmetric in 
the coordinates Xi and x 2 . We see that we are not justified in de- 
scribing the system in either one of these stationary states as con- 
sisting of the first oscillator in the state n x — 1 and the second in 
the state n 2 = 0, or the reverse. Instead, the wave functions 
ni = 1, n 2 = and ni = 0, n 2 = 1 contribute equally to each of 
the stationary states. It will be shown in Section 41c that if the 
perturbation is small we are justified in saying that there is reso- 
nance between these two states of motion analogous to classical 
resonance, one oscillator at a given time oscillating with large 
amplitude, corresponding to n x = 1, and at a later time with 
small amplitude, corresponding to n\ = 0. The frequency with 
which the oscillators interchange their oscillational states, that 
is, the frequency of the resonance, is found to be \/v Q} which 
is just equal to the separation of the two energy levels divided 
by h. This is also the frequency of the classical resonance 
(Eq. 41-5). 

In discussing the stationary states of the system of two inter- 
acting harmonic oscillators we have seen above that it is con- 
venient to make use of certain wave functions \p n (#i), etc. which 
are not correct wave functions for the system, the latter being 



XI-41b] THE RESONANCE PHENOMENON 321 

given by or approximated by linear combinations of the initially 
chosen functions, as found by perturbation or variation methods; 
and various points of analogy between this treatment and the 
classical treatment of the resonating system have been indicated 
(see also the following section) . In discussing more complicated 
systems it is often convenient to make use of similar methods of 
approximate solution of the wave equation, involving the forma- 
tion of linear combinations of certain initially chosen functions. 
The custom has arisen of describing this formation of linear 
combinations in certain cases as corresponding to resonance in 
the system. In a given stationary state the system is said to 
resonate among the states or structures corresponding to those 
initially chosen wave functions which contribute to the wave 
function for this stationary state, and the difference between 
the energy of the stationary state and the energy corresponding 
to the initially chosen wave functions is called resonance energy. 1 
It is evident that any perturbation treatment for a degenerate 
level in which the initial wave functions are not the correct 
zeroth-order wave functions might be described as involving 
the resonance phenomenon. Whether this description would 
be applied or not would depend on how important the initial 
wave functions seem to the investigator, or how convenient this 
description is in his discussion. 2 

The resonance phenomenon, restricted in classical mechanics 
to interacting harmonic oscillators, is of much greater importance 
in quantum mechanics, this being, indeed, one of the most striking 
differences between the old and the new mechanics. It arises, 
for example, whenever the system under discussion contains two 
or more identical particles, such as two electrons or two protons; 
and it is also convenient to make use of the terminology in 
describing the approximate treatment given the structure of 
polyatomic molecules. The significance of the phenomenon for 
many-electron atoms has been seen from the discussion of the 
structure of the helium atom given in Chapter VIII; it was there 
pointed out (Sec. 29a) that the splitting of levels due to the K 

1 There is no close classical analogue of resonance energy. 

2 The same arbitrariness enters in the use of the word resonance in describ- 
ing classical systems, inasmuch as if the interaction of the classical oscillators 
is increased the motion ultimately ceases to be even approximately repre- 
sented by the description of the first paragraph of Section 41a. 



322 PERTURBATION THEORY INVOLVING THE TIME [XI-41c 

integrals was given no satisfactory explanation until the develop- 
ment of the concept of quantum-mechanical resonance. The 
procedure which we have followed of delaying the discussion 
of resonance until after the complete treatment of the helium 
atom emphasizes the fact that the resonance phenomenon does 
not involve any new postulate or addition to the equations of 
wave mechanics but rather only a convenient method of classify- 
ing and correlating the results of wave mechanics and a basis for 
the development of a sound intuitive conception of the theory. 

41c. A Further Discussion of Resonance. — It is illuminating 
to apply the perturbation method of variation of constants in 
order to discuss the behavior of a resonating system. Let us 
consider a system for which we have two wave functions, say ^? A 
and V Bf corresponding to an energy level of the unperturbed 
system with two-fold degeneracy. These might, for example, 
correspond to the sets of quantum numbers rti — 1, n 2 =0 and 
ni = 0, n 2 = 1 for the system of two coupled harmonic oscillators 
treated in the previous section. If the perturbation were small, 
we could carry out an experiment at the time t = to determine 
whether the system is in state A or in state B; for example, we 
could determine the energy of the first oscillator with sufficient 
accuracy to answer this question. Let us assume that at the 
time t = the system is found to be in the state A . We now ask 
the following question: On carrying out the investigating experi- 
ment at a later time t, what is the probability that we would 
find that the system is in state A, and what is the probability 
that we would find it in state B? In answering this question 
we shall see that the physical interpretation of quantum-mechani- 
cal resonance is closely similar tp that of classical resonance. 

If the perturbation is small, with all the integrals U' mn {m ^ n) 
small compared with W Q n — W™ except H' AB and H BA (for which 
W° A = Wl), we may assume as an approximation that the 
quantities a m (t) remain equal to zero except for a A and a B . 
From Equation 39-6 we see that these two are given by the 
equations 

2*i, 

n l (41-10) 



6b = — ^(H'^cla + H' AA a B ),} 



in which we have taken H BB equal to H AA and H BA equal to H' AB 



XI-41c] THE RESONANCE PHENOMENON 323 

(the system being assumed to consist of two similar parts). 
The equations are easily solved by first forming their sum and 
difference. The solution which makes cu = 1 and a B = at 
t = is 

t A Al 



a A = e h cos ( — ^t h 

a B — —le h sm | 



m 



(41-11) 



The probabilities a*a A and a B a B of finding the system in state 
A and state B, respectively, at time t are hence 



a 2 a A — cosi 



\ h '( (41-12) 



We see that these probabilities vary harmonically between the 
values and 1. The period of a cycle (from a*a A = 1 to and 
back to 1 again) is seen to be h/2H' AB , and the frequency 2H' AB /h, 
this being, as stated in Section 416, just 1/A times the separation 
of the levels due to the perturbation. 

Let us now discuss in greater detail the sequence of conceptual 
experiments and calculations which leads us to the foregoing 
interpretation of our equations. Let us assume that we have a 
system composed of two coupled harmonic oscillators with 
coordinates Xi and x 2y respectively, such that we can at will (by 
throwing a switch, say) disengage the coupling, thus causing 
the two oscillators to be completely independent. Let us now 
assume that for a period of time previous to t = the oscillators 
are independent. During this period we carry out a set of two 
experiments consisting in separate measurements of the energy 
of the oscillators and in this way determine the stationary state 
of each oscillator. Suppose that by one such set of experiments 
we have found that the first oscillator is in the state n\ = 1 
and the second in the state n 2 = 0. The complete system is 
then in the physical situation which we have called state A in 
the above paragraphs, and so long as the system is left to itself 
it will remain in this state. 

Now let us switch in the coupling at the time t = 0, and then 
switch it out again at the time t = t'. We now, at times later 



324 PERTURBATION THEORY INVOLVING THE TIME [XI-41c 

than t', investigate the system to find what the values of the 
quantum numbers n\ and n 2 are. The result of this investigation 
will be the same, in a given case, no matter at what time later 
than t' the set of experiments is carried out, inasmuch as the 
two oscillators will remain in the definite stationary states in 
which they were left at time t r so long as the system is left 
unperturbed. 

This sequence of experiments can be repeated over and over, 
each time starting with the system in the state rii — 1, n 2 = 
and allowing the coupling to be operative for the length of time t r . 
In this way we can find experimentally the probability of finding 
the system in the various states n\ = 1, n 2 = 0;rii = 0, n 2 = 1; 
rii = 0, n 2 = 0; etc.; after the perturbation has been operative 
for the length of time t r . 

The same probabilities are given directly by our application of 
the method of variation of constants. The probability of 
transition to states of considerably different energy as the result 
of a small perturbation acting for a short time is very small, 
and we have neglected these transitions. Our calculation shows 
that the probability of finding the system in the state B depends 
on the value of t' in the way given by Equation 41-12, varying 
harmonically between the limits and 1. 

Now in case that we allow the coupling to be operative con- 
tinuously, the complete system can exist in various stationary 
states, which we can distinguish from one another by the measure- 
ment of the energy of the system. Two of these stationary 
states have energy values very close to the energy for the 
states rii = 1, n 2 = and n x = 0, n 2 = 1 of the system with 
the coupling removed. It is consequently natural for us to 
draw on the foregoing argument and to describe the coupled 
system in these stationary states as resonating between states 
A and B t with the resonance frequency 2H AB /h. 

Even when it is not possible to remove the coupling inter- 
action, it may be convenient to use this description. Thus in 
our discussion of the helium atom we found certain stationary 
states to be approximately represented by wave functions 
formed by linear combination of the wave functions ls(l) 2s(2) 
and 2s(l) ls(2). These we identify with states A and B above, 
saying that each electron resonates between a Is and a 2s orbit, 
the two electrons changing places with the frequency 1/A times 



XI-41c] THE RESONANCE PHENOMENON 325 

the separation of the energy levels ls2s l S and ls2s Z S. It is 
obvious that we cannot verify this experimentally, for three 
reasons: we cannot remove the coupling, we cannot distinguish 
electron 1 from electron 2, and the interaction is so large that 
our calculation (based on neglect of all other unperturbed states) 
is very far from accurate. These limitations to the physical 
verification of resonance must be borne in mind; but they need 
not prevent us from making use of the nomenclature whenever 
it is convenient (as it often is in the discussion of molecular 
structure given in the following chapter). 



CHAPTER XII 
THE STRUCTURE OF SIMPLE MOLECULES 

Of the various applications of wave mechanics to specific 
problems which have been made in the decade since its origin, 
probably the most satisfying to the chemist are the quantitatively 
successful calculations regarding the structure of very simple 
molecules. These calculations show that we now have at hand 
a theory which can be confidently applied to problems of molec- 
ular structure. They provide us with a sound conception of the 
interactions causing atoms to be held together in a stable mole- 
cule, enabling us to develop a reliable intuitive picture of the 
chemical bond. To a considerable extent the contribution of 
wave mechanics to our understanding of the nature of the 
chemical bond has consisted in the independent justification of 
postulates previously developed from chemical arguments, and 
in the removal of their indefinite character. In addition, 
wave-mechanical arguments have led to the development of many 
essentially new ideas regarding the chemical bond, such as the 
three-electron bond, the increase in stability of molecules by 
resonance among several electronic structures, and the hybridi- 
zation of one-electron orbit als in bond formation. Some of 
these topics will be discussed in this chapter and the following 
one. 

In Sections 42 and 43 we shall describe the accurate and 
reliable wave-mechanical treatments which have been given the 
hydrogen molecule-ion and hydrogen molecule. These treat- 
ments are necessarily rather complicated. In order to throw 
further light on the interactions involved in the formation of 
these molecules, we shall preface the accurate treatments by a 
discussion of various less exact treatments. The helium mole- 
cule-ion, He£, will be treated in Section 44, followed in Section 45 
by a general discussion of the properties of the one-electron bond, 
the electron-pair bond, and the three-electron bond. 

326 



XII-42a] THE HYDROGEN MOLECULE-ION 327 

42. THE HYDROGEN MOLECULE-ION 

The simplest of all molecules is the hydrogen molecule-ion, Hf , 
composed of two hydrogen nuclei and one electron. This mole- 
cule was one of the stumbling blocks for the old quantum theory, 
for, like the helium atom, it permitted the treatment to be carried 
through (by Pauli 1 tod Niessen 2 ) to give results in disagreement 
with experiment. It was accordingly very satisfying that within 
a year after the development of wave mechanics a discussion 
of the normal state of the hydrogen molecule-ion in complete 
agreement with experiment was carried out by Burrau by 
numerical integration of the wave equation. This treatment, 
together with somewhat more refined treatments due to Hylleraas 




A r AB b 

Fig. 42—1. — Coordinates used for the hydrogen molecule-ion. 

and Jaff6, is described in Section 42c. Somewhat simpler and 
less accurate methods are described in Sections 42a and 426, 
for the sake of the ease with which they can be interpreted. 

42a. A Very Simple Discussion. 3 — Following the discussion of 
Section 34, the first step in the treatment of the complete wave 
equation is the solution of the wave equation for the electron 
alone in the field of two stationary nuclei. Using the symbols 
of Figure 42-1, the electronic wave equation is 

in which v 2 refers to the three coordinates of the electron and m 
is the mass of the electron. 4 

1 W. Pauli, Ann. d. Phys. 68, 177 (1922). 

2 K. F. Niessen, Dissertation, Utrecht, 1922. 
8 L. Pauling, Chem. Rev. 5, 173 (1928). 

4 We have included the mutual energy of the two nuclei e*/HAB in this 
equation. This is not necessary, inasmuch as the term appears unchanged 
in the final expression for W, and the same result would be obtained by 
omitting it in this equation and adding it later. 



328 



THE STRUCTURE OF SIMPLE MOLECULES [XII-42a 



If Tab is very large, the normal state of the system has the 
energy value W = Wn = —Rhc, the corresponding wave 
functions being u Ua or ^i v hydrogen-atom wave functions 
about nucleus A or nucleus B (Sec. 21), or any two independent 
linear combinations of these. In other words, for large values 
of Tab the system in its normal state is composed of a hydrogen 
ion A and a normal hydrogen atom B or of a normal hydrogen 
atom A and a hydrogen ion B. 

This suggests that as a simple variation treatment of the 
system for smaller values of r AB we make use of the same wave 
functions Uu a and u u , forming the linear combinations given by 
solution of the secular equation as discussed in Section 26<i. 
The secular equation is 

H AA - W H AB - AW 
H BA - AW Ebb - W 



= 0, 



(42-2) 



in which 



and 



II a a = jUuHuudr, 
Hab = JuujHuujdr, 



A = JuuUudr. 

A represents the lack of orthogonality of u Ua and Uu . Because 
of the equivalence of the two functions, the relations H A a = H B b 
and Hab = H B a hold. The solutions of the secular equation are 
hence 

Haa + Hab 



and 



Ws 



W A = 



1+* 



Haa — Ha 

1 - A 



These correspond respectively to the wave functions 



and 






(42-3) 
(42-4) 

(42-5) 
(42-6) 



The subscripts S and ^4 represent the words symmetric and 
antisymmetric, respectively (Sec. 29a) ; the wave function p 3 is 



XII-42a] THE HYDROGEN MOLECULE-ION 329 

symmetric in the positional coordinates of the two nuclei A and B, 
and \j/A is antisymmetric in these coordinates. 
Introducing Wh by use of the equation 

h 2 e 2 

8ir'mo A Ta A A 

(which is the wave equation for u u ), we obtain for the integral 
Haa the expression 

Haa = (u u ( Wh - e l + y\ u dr = W H + J + ~ 5 > (42-7) 

in which 

In this expression we have introduced in place of r AB the variable 

D = r -^- (42-9) 

do 

Hba and H A b are similarly given by the expression 

Hba = (uuIWh - - + —\udr = ATFh + K + ~, 

J *\ r B Tab) a CloD 

(42-10) 
in which A is the orthogonality integral, with the value 

A = e- D (l +D + y 3 D>), (42-11) 

and K is the integral 

K = Juu B (-f)u u dT = -jf D H + D). (42-12) 

It is seen that J represents the Coulomb interaction of an 
electron in a Is orbital on nucleus A with nucleus B. K may 
be called a resonance or exchange integral, since both functions 
Uu a and Uu B occur in it. 

Introducing these values in Equations 42-3 and 42-4, we 
obtain 

and 

w " w ' + £> + T^f • < 42 - 14 > 



330 



THE STRUCTURE OF SIMPLE MOLECULES [XII-42a 



Curves showing these two quantities as functions of r AB are given 
in Figure 42-2. It is seen that yps corresponds to attraction, with 
the formation of a stable molecule-ion, whereas \f/A corresponds to 
repulsion at all distances. There is rough agreement between 
observed properties of the hydrogen molecule-ion in its normal 
state and the values calculated in this simple way. The dis- 
sociation energy, calculated to be 1.77 v.e., is actually 2.78 v.e., 
and the equilibrium value of r AB , calculated as 1.32 A, is observed 
to be 1.06 A. 
The nature of the interactions involved in the formation of 

this stable molecule (with a one- 
electron bond) is clarified by the 
discussion of a hypothetical case. 
Let us assume that our system is 
composed of a hydrogen atom 
A and a hydrogen ion B, and 
that even for small values of r AB 
the electron remains attached to 
nucleus A, the wave function 
being u u . The energy of the 



•Q8 

-0.9 

t 
W-1.0 

11 

•1.2 











A 








vN 


























r o 







I 2 5 4 5 6 

TAB/dQ— * 

Fig. 42-2. — Energy curves for the 
hydrogen molecule-ion (in units 
eV2ao), calculated for undistorted system Would then be H A a, and 
hydrogen atom wave functions. ^ difference be tween this and 



W H , namely 



H l+ i> 



would be the energy of interaction 



of a normal hydrogen atom and a hydrogen ion. The curve 
representing this energy function, which before the discovery 
of the resonance phenomenon was supposed to correspond io 
the hydrogen molecule-ion, is shown in Figure 42-2 with the 
symbol N. It is seen that it does not correspond to the formation 
of a stable bond but instead to repulsion at all distances. The 
difference between this curve and the other two is that in this 
case we have neglected the resonance of the electron between 
the two nuclei A and B. It is this resonance which causes 
the actual hydrogen molecule-ion to be stable — the energy of the 
one-electron bond is in the main the energy of resonance of the 
electron between the two nuclei. (Other interactions, such as 
polarization of the atom in the field of the ion, also contribute 
to some extent to the stability of the bond. An attempt to 
anawer the question of the magnitude of this contribution will 
be given in the next section.) 



XII-42bJ THE HYDROGEN MOLECULE-ION 331 

It is seen from the figure that the resonance interaction sets 
in at considerably larger distances than the Coulomb interaction 
of atom and ion. This results from the exponential factor 
e~ 2D in Haa, as compared with e~ D in the resonance integral K. 
For values of r AB larger than 2 A the energy functions Ws and W a 
are closely approximated by the values Wh + K and Wh — K, 
respectively. In accordance with the argument of Section 
416, the resonance energy ±K corresponds to the electron's 
jumping back and forth between the nuclei with the frequency 
2K/h. 

Problem 42-1. Verify the expressions given for Haa, Hab, and A in 
Equations 42-7 to 42-12. 

42b. Other Simple Variation Treatments. — We can easily 
improve the preceding treatment by introducing an effective 
nuclear charge Z'e in the hydrogenlike Is wave functions u u 
and u u . This was done by Finkelstein and Horowitz. 1 On 
minimizing the energy Ws relative to Z' for various values of 
r AB , they obtained a curve for Ws similar to that of Figure 42-2, 
but with a lower minimum displaced somewhat to the left. They 
found for the equilibrium value of r AB the value 1.06 A, in com- 
plete agreement with experiment. The value of the effective 
atomic number Z f at this point is 1.228, and the energy of the sys- 
tem (neglecting oscillational and rotational energy) is — 15.78 v.e., 
as compared with the correct value —16.31 v.e.; the value 
of the dissociation energy D e = 2.25 v.e. differing from the cor- 
rect value 2.78 v.e. by 0.53 v.e. The variation of the effective 
atomic number from the value 1 has thus reduced the error 
by one-half. 

The energy of the bond for this function too is essentially 
resonance energy. Dickinson 2 introduced an additional term, 
dependent on two additional parameters, in order to take 
polarization into account. He wrote for the (not yet normalized) 
variation function 

* = u Ua (Z') + uuJLZ') + <t{u 2Pa (Z") + u 2Pb (Z")}, 

in which the first two terms represent as before Is hydrogenlike 
wave functions with effective nuclear charge Z'e and the remain- 

1 B. N. Finkelstein and G. E. Horowitz, Z.f. Phys. 48, 118 (1928). 
* B. N. Dickinson, /. Chem. Phys, 1, 317 (1933). 



332 THE STRUCTURE OF SIMPLE MOLECULES [XII-42b 



u **a =777^(77y""T: r ^ ^os#, 



ing two terms functions such as 2p z as described in Section 21, 

'Z"\* Z" -f'-'rx 

4V27r\ «o / a 

in which # is taken relative toaz axis extending from nucleus 
A toward nucleus B (and the reverse for u 2Pb ). The parameter 
a determines the extent to which these functions enter. The 
interpretation of the effect of these functions as representing 
polarization of one atom by the other follows from their nature. 
The function Uu A + <tu 2Pa differs from u Ua by a positive amount 
on the side nearer B and a negative amount on tbe farther side, 
in this way being concentrated toward B in the way expected for 
polarization. 1 

On minimizing the energy relative to the three parameters and 
to v% By Dickinson found for the equilibrium distance the value 
1.06 A, and for the energy —16.26 v.e., the parameters having 
the values Z f = 1.247, Z" = 2.868, and a = 0.145. 2 The 
energy calculated for this function differs by only 0.05 v.e. 
from the correct value, so that we may say, speaking somewhat 
roughly, that the energy of the one-electron bond is due almost 
entirely to resonance of the electron between the two nuclei and 
to polarization of the hydrogen atom in the field of the hydrogen 
ion, with resonance making the greater contribution (about 
2.25 v.e., as given by Finkelstein and Horowitz's function) and 
polarization the smaller (about 0.5 v.e.). 

It was found by Guillemin and Zener 3 that another variation 
function containing only two parameters provides a very good 
value for the energy, within 0.01 v.e. of the correct value, the 
equilibrium separation of the nuclei being 1.06 A, as for all 
functions discussed except the simple one of the preceding 
section. This function is 

asp ao _L_ /> aog ao 

1 The introduction of such a function to take care of polarization was 
first made (for the hydrogen molecule) by N. Rosen, Phys. Rev. 38, 2099 
(1931). 

1 It will be noted that Z" is approximately twice Z'. Dickinson found 
that the error in the energy is changed only by 0.02 v.e. by placing Z" 
equal to 2Z', the best values of the parameters then being Z' = 1.254, 
a = 0.1605. 

* V. Guillemin, Jr., and C. Zener, Proc. Nat. Acad. Set. 16, 314 (1929). 



XII-42c] THE HYDROGEN MOLECULE-ION 333 

the best values of the parameters being Z f = 1.13, Z" = 0.23. 
The interpretation of this function is not obvious; we might say, 
however, that each of the two terms of the function represents 
a polarized hydrogen atom, the first term, for example, being 
large only in the neighborhood of nucleus A, and being there 

polarized in the direction of nucleus B by the factor e ao 

-z' r -± 
multiplying the hydrogenlike function e ao , the entire wave 

function then differing from Dickinson's mainly in the way in 

which the polarization is introduced. The value of the principal 

effective atomic number Z' = 1.13 is somewhat smaller than 

Dickinson's value 1.247. 

A still more simple variation function giving better results 

has been recently found by James. 1 This function is 

<r«(l + of), 

in which £ and 77 are the confocal elliptic coordinates defined in 
the following section (Eq. 42-15), and 8 and c are parameters? 
with best values 8 = 1.35 and c = 0.448. The value of the 
dissociation energy given by this function is D e — 2,112 v.e., 
the correct value being 2.777 v.e. 

42c. The. Separation and Solution of the Wave Equation. — It 
was pointed out by Burrau 2 that the wave equation for the 
hydrogen molecule-ion, Equation 42-1, is separable in confocal 
elliptic coordinates £ and rj and the azimuthal angle <p. The 
coordinates £ and 77 are given by the equations 

t r A + r B \ 

f = — T 7 

AB } (42-15) 

Ta — r B I 

*? = — r ) 

Tab I 

On introduction of these coordinates (for which the Laplace 
operator is given in Appendix IV), the wave equation becomes 

¥%&{%& - ,■> + £«}* - 0, (42-16) 

1 H. M. James, private communication to the authors. 

1 0yvind Burrau, Det. Kgl. Danske Vid. Selskab. 7, 1 (1927). 



334 THE STRUCTURE OF SIMPLE MOLECULES [XII-42c 

in which we have made use of the relation 

f! +^! = 4e 2 ^ 
Ta r B r AB (Z 2 - v 2 ) 

and have multiplied through by • , — — • The quantity 
W f , given by 



pp = W - — , (42-17) 

Tab 

is the energy of the electron in the field of the two nuclei, the 
mutual energy of the two nuclei being added to this to give the 
total energy W. 

It is seen that on replacing i^(£, ?j, <p) by the product function 

*({, u, *>) = E({)H(i,)*(?) (42-18) 

this equation is separable 1 into the three differential equations 

B = - m2 *' (42_19) 



|{(l-^} + (x^- r ^-.)H=0, (42-20) 

|{« 2 " d| } + (~^ + *>* ~ w^j + ") s = °' (42 " 21) 



and 



in which 



and 



X = ,^ (42-22) 



D = ^2. (42-23) 

a 

The range of the variable J is from 1 to °o , and of rj from — 1 to 
+ 1. The surfaces £ = constant are confocal ellipsoids of revolu- 
tion, with the nuclei at the foci, and the surfaces t\ = constant 
are confocal hyperboloids. The parameters m, X, and /z must 
assume characteristic values in order that the equations possess 
acceptable solutions. The familiar <p equation possesses such 
solutions for m = 0, ±1, ±2, • • • . The subsequent procedure 
of solution consists in finding the relation which must exist 

1 The equation is also separable for the case that the two nuclei have 
different charges. 



XII-42cl 



THE HYDROGEN MOLECULE-ION 



335 



-08 



between X and /x in order that the t\ equation possess a satis- 
factory solution, and, using this relation, in then finding from the 
f equation the characteristic values of X and hence of the energy. 

This procedure was carried out for the normal state of the 
hydrogen molecule-ion by Burrau in 1927 by numerical integra- 
tion of the £ and rj equations. More accurate treatments have 
since been given by Hylleraas 1 and by Jaflte. 2 (The simple 
treatment of Guillemin and Zener, described in the preceding 
fection, approaches Burrau's 
in accuracy.) We shall not 
describe these treatments in 
detail but shall give a brief dis- 
cussion of one of them (that 
of Hylleraas) after first pre- 
senting the results. 

The energy values calculated 
by the three authors are given 
in Table 42-1 and shown graph- 
ically in Figure 42-3. It is seen 
that the curve is qualitatively 
similar to that given by the very 



-0.9 



wt 



-1.0 



-1.2 



-1.3 



1 



r AB/a 



ix A , ro x. a* FlG ' 42-3.— The energy of the 

Simple treatment Ot bection 42a normal hydrogen molecule-ion (in units 

(Fig. 42-2). The three treat- e2/2ao > as a function of tab. 
ments agree in giving for the equilibrium value 3 of r AB 2.00 a or 
1.06 A, as was found for the variation functions of the preceding 
section also. This is in complete agreement with the band- 
spectral value. Spectroscopic data have not been obtained for 
the hydrogen molecule-ion itself but rather for various excited 
states of the hydrogen molecule. It is believed that these are 
states involving a normal hydrogen molecule-ion as core, with a 
highly excited outer electron in a large orbit, having little effect 

1 E. A. Hylleraas, Z. f. Phys. 71, 739 (1931). 

2 G. Jaffe, Z. f. Phys. 87, 535 (1934). 

3 The average value of tab for various oscillational states as determined 
from band-spectral data is found to be a function of the vibrational quantum 
number v, usually increasing somewhat with increasing v. The value 
for v = is represented by the symbol r , and the extrapolated value corre- 
sponding to the minimum of the electronic energy function by the symbol 
r 6 . The vibrational frequencies are similarly represented by *> and v 4 
(or by wo and « a , which have found favor with band spectroscopists) and 
the energies of dissociation by Do and D«. 



336 



THE STRUCTURE OF SIMPLE MOLECULES [XII -42c 



on the potential function for the nuclei; this belief being supported 
by the constancy of the values of r e and v e (the oscillational 
frequency) shown by them. The values of r e were extrapolated 
by Birge 1 and Richardson 2 to give 1.06 A for the molecule-ion. 

Table 42-1. — Electronic Energy Values for the Hydrogen 
Molecule-ion 





Wiij (in units Rhc) 


tab/cio 


Burrau 


Guillemin 
and Zener 


Hylleraas 


Jafte 



0.5 


00 

-0.896 

-1 195* 
-1 204 
-1.198* 

-1.000 


00 


00 

0.5302 
-0.9046 
-1.0826 
-1.1644 
-1.1980 
- 1 . 20527 
- 1 . 1998 
-1.1878 
-1.1716 
-1.1551 
-1.0000 


00 

5300 


1.0 
1.25 


-0.903 


-0.9035 


1.5 






1.75 
2.00 
2.25 
2 5 


-1.198f 
-1.205 
— 1 . 197f 


- 1 . 20528 


2.75 
3.0 

00 


-1 154 
-1 000 


-1.1544 
-1 0000 



* Interpolated between adjacent values calculated by Burrau, who estimated his accu- 
racy in the neighborhood of the minimum as ± 0.002 Rhc. 
t Interpolated values. 

The value — 1.20528jR/ic for the energy of the molecule-ion is 
also substantiated by experiment; the discussion of this com- 
parison is closely connected with that for the hydrogen molecule, 
and we shall postpone it to Section 43d. The behavior of the 
minimum, however, can be compared with experiment by way 
of the vibrational energy levels. By matching a Morse curve 
to his calculated points and applying Morse's theory (Sec. 35d), 
Hylleraas found for the energy of the molecule ion in successive 
vibrational levels given by the quantum number v the expression 
W v = -1.20527 + 0.0206(y + Y 2 ) - 0.000510 + y 2 )\ (42-24) 
in units Rhc. This agrees excellently with the expressions 
obtained by Birge 1 and Richardson 2 by extrapolation of the 
observed vibrational levels for excited states of the hydrogen 

1 R. T. Birge, Proc. Nat. Acad. Set. 14, 12 (1928). 

*0. W. Richardson, Trans. Faraday Soc. 26, 686 (1929). 



XII-42C 



THE HYDROGEN MOLECULE-ION 



337 



molecule, their coefficients in these units being 0.0208 and 
-0.00056, and 0.0210 and -0.00055, respectively. 

The value W e = — 1. 20527 R H hc corresponds to an electronic 
energy of the normal hydrogen molecule-ion of —16.3073 v.e. 
(using Rnhc = 13.5300 v.e.) and an electronic dissociation energy 
into H + H+ of D e = 2.7773 v.e., this value being shown to be 
accurate to 0.0001 v.e. by the agreement between the calculations 
of Hylleraas and Jaff6. The value of Z> , the dissociation energy 
of the molecule-ion in its lowest vibrational state, differs from 
this by the correction terms given in Equation 42-24. These 




Fig. 42-4. — The electron distribution function for the normal hydrogen 
molecule-ion (Burrati). The upper curve shows the value of the function along 
the line passing through the two nuclei, and the lower figure shows contour 
lines for values 0.9, 0.8, • • • ,0.1 times the maximum value. 

terms are not known so accurately, either theoretically or 
experimentally. Hylleraas's values lead to a correction of 
0.138 v.e., Birge's to 0.139 v.e., and Richardson's to 0.140 v.e. 
If we accept the theoretical value 0.138 ± 0.002 v.e. we obtain 

Do = 2.639 ± 0.002 v.e. 

as the value of the dissociation energy of the normal hydrogen 
molecule-ion. 

The wave function for the normal molecule-ion as evaluated by 
Burrau corresponds to the electron distribution function repre- 
sented by Figure 42-4. It is seen that the distribution is 
closely concentrated about the line between the two nuclei, 
the electron remaining most of the time in this region. 

Let us now return to a brief discussion of one of the accurate 
treatments of this system, that of Hylleraas, which illustrates 



338 THE STRUCTURE OF SIMPLE MOLECULES [XII-42c 

the method of approximate solution of the wave equation dis- 
cussed in Section 27a. 

The variable r; extends through the range —1 to +1, which 
is the range traversed by the argument z = cos & of the associated 
Legendre functions Pi mI of Section 19. With Hylleraas we 
expand the function H(r/) in terms of these functions, writing 

00 

HW = X c i p \ m ^), (42-25) 

i-IH 

in which the coefficients c t are constants. Substituting this 
expression in Equation 42-20, and simplifying with the aid of the 
differential equation satisfied by the associated Legendre func- 
tions, Equation 19-9, we obtain the equation 

00 

2 c,{W -n-l(]i + IJIWn) = 0. (42-26) 

i-|m| 

We can eliminate the factor j; 2 by the use of the recursion formula 



(I - \m\ + l)(l - \m\ + 2) , 
(21 + 1)(2I + 3) 



„»PMM = St ~ m ^ ill!. ~ l w ' t *> p\»>i 



( q _ \ m \ + m + H + 1) (I - \m\)(l + |mf) | p,„ 
+ \ (2 + l)(H + 3) + (21 - 1)(2J + 1) f ' 

I (l + \m\)(l + \m\-l) 
+ (21 - l)(2l + 1) ^" 2J (42 ^ 27) 

which is easily obtained by successive application of the ordinary 
recursion formula 19-16. On introducing this in Equation 
42-26, it becomes a simple series in the functions Pi ml M with 
coefficients independent of r\. Because of the orthogonality of 
these functions, their coefficients must vanish independently in 
order that the sum vanish (Sec. 22). This gives the condition 

(l-\m\-l)(l-\m\\ „ , r/ (j ~ H + W + H + 1) 

(21 - 3) (a - i) l ' 2 + [\ (a + i)(a + 3) 

which is a three-term recursion formula in the coefficients Ci. 



XH-42c] 



THE HYDROGEN MOLECULE-ION 



339 



We now consider the set of equations 42-28 for different values 
of I as a set of simultaneous linear homogeneous equations in the 
unknown quantities ci. In order that the set may possess a non- 
trivial solution, the determinant formed by the coefficients of 
the Ci'b must vanish. This gives a determinantal equation 
involving X and n, from which we determine the relation between 
them. 

We are interested in the normal state of the system, with m = 
and I even. The determinantal equation for this case is 



X — n 






11 

21 



15 
X - 6 - m 

35 X 













39 

77 



X - 20 - /x 




= 0. 

(42-29) 



The only non-vanishing terms are in the principal diagonal and 
the immediately adjacent diagonals. As a rough approximation 
(to the first degree in X) we can neglect the adjacent diagonals; 
the roots of the equation are then \i = J^X, m = x Ki^ — 6, 
M = 3 ^77^ ~~ 20, etc. We are interested in the first of these. 
In order to obtain it more accurately, we solve the equation again, 
including the first two non-diagonal terms, and replacing \x in 
the second diagonal term by J^X. This equation, 



= 0, 



M = MX + K35X 2 + HsosX*, 
in which powers of X higher than the third are neglected. Hyller- 
aas carried the procedure one step farther, obtaining 

M = y z \ + % 35 x 2 + ^ 5 05* 3 - 0.000013X 4 - 0.0000028X 5 . 

This equation expresses the functional dependency of m on X 
for the normal state, as determined by the 77 equation. The 
next step is to introduce this in the £ equation, eliminating /x, 
and then to solve this equation to obtain the characteristic 





3X-M 




£ 




1* 


21 X 


-6-ix 


has the solution 









340 THE STRUCTURE OF SIMPLE MOLECULES [XII-43a 

values of X and hence of the energy as a function of r AB . Because 
of their more difficult character, we shall not discuss the methods 
of solution of this equation given by Hylleraas and Jafife. 

42d. Excited States of the Hydrogen Molecule-ion. — We 
have discussed (Sec. 42a) one of the excited electronic states of 
the hydrogen molecule-ion, with a nuclear-antisymmetric wave 
function formed from normal hydrogen-atom functions. This 
is not a stable state of the molecule-ion, inasmuch as the potential 
function for the nuclei does not show a minimum. 

Calculations of potential functions for other excited states, 
many of which correspond to stable states of the molecule-ion, 
have been made by various investigators, 1 among whom 
Teller, Hylleraas, and JafT6 deserve especial mention. 

43. THE HYDROGEN MOLECULE 

43a. The Treatment of Heitler and London. — The following 
simple treatment of the hydrogen molecule (closely similar to 
that of the hydrogen molecule-ion described in Section 42a) 
does not differ essentially from that given by Heitler and London 2 
in 1927, which marked the inception (except for Burrau's earlier 
paper on the molecule-ion) of the application of wave mechanics 
to problems of molecular structure and valence theory. Heitler 
and London's work must be considered as the greatest single contri- 
bution to the clarification of the chemist's conception of valence 
which has been made since G. N. Lewis's suggestion in 1916 that 
the chemical bond between two atoms consists of a pair of 
electrons held jointly by the two atoms. 

Let us first consider our problem with neglect of the spin of 
the electrons, which we shall then discuss toward the end of the 
section. The system comprises two hydrogen nuclei, A and B, 
and two electrons, whose coordinates we shall designate by the 
symbols 1 and 2. Using the nomenclature of Figure 43-1, the 
wave equation for the two electrons corresponding to fixed posi- 
tions of the two nuclei is 

1 P. M. Morse and E. C. G. Stueckelberg, Phys. Rev. 33, 932 (1929); 
E. A. Hylleraas, Z.J. Phys. 61, 150 (1928); 71, 739 (1931); J. E. Lennard- 
Jones, Trans. Faraday Soc. 24, 668 (1929); E. Teller, Z. J. Phys. 61, 458 
(1930); G. Jaffe, Z.J. Phys. 87, 535 (1934). 

1 W. Heitler and F. London, Z. J. Phys. 44, 455 (1927). 



XII -43a] 



THE HYDROGEN MOLECULE 



341 



V?* + V?* + 



87r 2 m J 



Tax r B \ r A i r B 2 f*i 2 



£}<=»• 



(43-1) 



For very large values of r AB we know that in its normal state the 
system consists of two normal hydrogen atoms. Its wave func- 
tions (the state having two-fold degeneracy) are then u u (1) 
Uu b (2) and u Ub (1) u Ua (2) or any two independent linear com- 
binations of these two (the wave function u Ua (1) representing a 
hydrogenlike Is wave function for electron 1 about nucleus A, 




A r AB B 

Fig. 43-1. — Coordinates used for the hydrogen molecule, represented diagram- 

matically. 



etc., as given in Section 21). This suggests that for smaller 
values of t A b we use as variation function a linear combination of 
these two product functions. We find as the secular equation 
corresponding to this linear variation function (Sec. 26c?) 



Hn-W #i n - A 2 Tf 



= 0, 



(43-2) 



in which 



and 



with 



Hi i = ffxpiHfadndrz, 
Hi ii = Jj\l/iH\l/ndTidT 2j 

A 2 = JfhhidTidT2, 

h = Uu A (l)uu a (2) and ^ n = 1^(1)^(2). 



It is seen that A is the orthogonality integral introduced in Section 
42a, and given by Equation 42-11. With Hi i = Hu n and 
Hi ii = Hn i, the equation can be immediately solved to give 



342 THE STRUCTURE OF SIMPLE MOLECULES [XII-48a 



and 



w _ Hi i + Hi ii ( , Q „v 

Ws 1 + A » ( 3 ~ 3 ' 



TI7 Hi l ~~ Hi U /AO A\ 

Wa = — i - a 2 ^ (43-4) 



corresponding to the wave functions 

*, = —L^ [uu A (l)u l§B (2) + u Ub (1)uu a (2) } (43-5) 
and 

+ A = -^={^(1)^(2) - u Ub (1)uu a (2)}. (43-6) 

^s is symmetric in the positional coordinates of the two electrons 
and also in the positional coordinates of the two nuclei, whereas 
\(/a is antisymmetric in both of these sets of coordinates. 
On evaluation we find for Hi i the expression 

ft i = f f «i..(D«,.,(2)(W„ -*--*-+*.+ *-) 

J J \ r B \ r A 2 r 12 Tab/ 

Wi^(l)wi. fl (2)dr 1 dr 2 

= 2W H + 2J + J' + — , (43-7) 

Tab 



in which J is the integral of Equation 42-8 and J' is 

="5{b-«-Kb + T + I c +H}' (43 - 8) 

with D as before equal to TabAo. 

Similarly we find for Hi n the expression 

ft„ = JJ^(l) M ,,(2)(2^-£-^ + £ 

— JWi^(l)u 1 . A (2)dridr 1 

- 2A 2 TF* + 2A# + £' + A 2 —, (43-9) 
in which if is the integral of Equation 42-12 and K' is 



Xn-48a] THE HYDROGEN MOLECULE 343 

,(1^(2)11^(1^(2) 






-£[-«-■< -¥ + f o+3D,+ H 

+ ~{A 8 (7 + log D) + A'*Ei(-4D) - 2AA , ^(~2D)} 1, 

(43-10) 
in which y =0.5772 • • • is Euler's constant and 

A' - e D (l - D + HD*). 
Ei is the function known as the integral logarithm. 1 (The 
integral K' was first evaluated by Sugiura, 2 after Heitler and 
London had developed an approximate expression for it.) J' 
represents the Coulomb interaction of an electron in a Is orbital 
on nucleus A with an electron in a Is orbital on nucleus 
By and K f is the corresponding resonance or exchange integral. 
Substituting these values in Equations 43-3 and 43-4, we 
obtain 



Ws - 2W " + 7T B + «x7 T (43-iD 



and 



e 



2J + J' 


+ 2AK + K' 


1 


+ A 2 




2J + J' 


- 2AK- 


-K' 



Wa = 2Wh + fz> + i -~ — -• ( 43 - 12 > 

Curves representing Ws and Wa as functions of r AB are shown in 
Figure 43-2. It is seen that Wa corresponds to repulsion at all 
distances, there being no equilibrium position of the nuclei. 
The curve for Wa corresponds to attraction of the two hydrogen 
atoms with the formation of a stable molecule, the equilibrium 
value calculated for t A b being 0.80 A, in rough agreement with 
the experimental value 0.740 A, The energy of dissociation of 
the molecule into atoms (neglecting the vibrational energy of the 
nuclei) is calculated to be 3.14 v.e., a value somewhat smaller 
than the correct value 4.72 v.e. The curvature of the potential 
function near its minimum corresponds to a vibrational frequency 
for the nuclei of 4800 cm -1 , the band-spectral value being 
4317.9 cm- 1 . 

It is seen that even this very simple treatment of the problem 
leads to results in approximate agreement with experiment. 

1 See, for example, Jahnke and Emde, << Funktionentafeln. , ' 
* Y. Suqiura, Z. /. Phy8. 46, 484 (1927). 



344 



THE STRUCTURE OF SIMPLE MOLECULES [XII-43a 



It may be mentioned that the accuracy of the energy calculation 
is greater than appears from the values quoted for D ej inasmuch 
as the energy of the electrons in the field of the two nuclei 
(differing from Wa by the term e 2 /r AB ) at r AB = 1.5a is calculated 
to be 2W H — 18.1 v.e., and the error of 1.5 v.e. is thus only a 
few per cent of the total electronic interaction energy. 

It is interesting and clarifying for this system also (as for 
the hydrogen molecule-ion) to discuss the energy function for a 
hypothetical case. {Let us suppose that the wave function for 
the system were fa = ^i^(l) u Ub (2) alone. The energy of the 
system would then be Hi i, which is shown as curve N in Figure 

43-2. (it is seen that this curve 



-1.4 
-1.6 

w 

-2.0 
-2.2 
-24 















1 




yA 








i" 














v" 





















I 



gives only a small attraction 
between the two atoms, with a 
bond energy at equilibrium only 
a few per cent of the observed 
value. The wave function fa 
differs from this function in the 
interchange of the coordinates 
of the electrons, and we conse- 
quently say that the energy 
r AB/a ~+* of the bond in the hydrogen 

Fio. 43-2.— Energy curves for the molecule is in the main reso- 

hydrogen molecule (in units e 2 /2ao). ~ * . « i 

nance or interchange energy. 
So far we have not taken into consideration the spins of the 
electrons. On doing this we find, exactly as for the helium 
atom, that in order to make the complete wave functions anti- 
symmetric in the electrons, as required by Pauli's principle, the 
orbital wave functions must be multiplied by suitably chosen 
spin functions, becoming 

1 



and 



fa~{a(l)P(2) -j3(l)a(2)} 

V2 

^•«(l)a(2), 
^~{a(l)«2)+j9(l)a(2)}, 

1^.0(1)0(2). 
There are hence three repulsive states A for one attractive 
state S; the chance is % that two normal hydrogen atoms 



XII-43b] 



THE HYDROGEN MOLECULE 



345 



will interact with one another in the way corresponding to the 
formation of a stable molecule. It is seen that the normal state 
of the hydrogen molecule is a singlet state, the spins of the 
two electrons being opposed, whereas the repulsive state A is p 
triplet state. 

43b, Other Simple Variation Treatments. — The simple step 
of introducing an effective nuclear charge Z'e in the Is hydrogen- 
like wave functions of 43-5 was taken by Wang, 1 who found that 
this improved the energy somewhat, giving D e = 3.76 v.e., 
and that it brought the equilibrium internuclear distance r e 
down to 0.76 A, only slightly 
greater than the experimental 
value 0.740 A. The effective 
nuclear charge at the equilib- 
rium distance was found to be 
Z e = 1.1666. 

There exists the possibility 
that wave functions correspond- 
ing to the ionic structures H~H + 
and H + H~ might also make 



-5ve 

10 

15 











































1 



5A 



r AB"*^ 
Fig. 43-3.— The mutual Coulomb 
an appreciable Contribution to energy of two ions with charges -He 
ji r x • r ii and — e as a function of txb> 

the wave function tor the nor- 
mal state of the molecule. These ionic functions arewi 8 (1) u u (2) 
and Uu B (l) Uu B (2), the corresponding spin function allowed 

by Pauli's principle being -— { a (l) 0(2) - 0(1) a(2)}, as for 

V2 
fa. It is true that for large values of t A b the energy of the ionic 
functions is 12.82 v.e. greater than that for the atomic functions, 
this being the difference of the ionization potential and the 
electron affinity (Sec. 29c) of hydrogen; but, as r AB is decreased, 
the Coulomb interaction of the two ions causes the energy for the 
ionic functions to decrease rapidly, as shown in Figure 43-3, 
the difference of 12.82 v.e. being counteracted at 1.12 A. This 
rough calculation suggests that the bond in the hydrogen molecule 
may have considerable ionic character, the structures H~H + and 
H+H~ of course contributing equally. The wave function 

Uu A (l) u Ub (2) + Uu B (l) u Ua (2) + c{u,^(l) Uu A (2) + 

Uu B (l) Uu B (2)} (43-13) 
1 S. C. Wang, Phys. Rev. 31, 579 (1928). 



346 



THE STRUCTURE OF SIMPLE MOLECULES [XII-43b 



-2.16 



was considered by Weinbaum, 1 using an effective nuclear charge 
Z'e in all the Is hydrogenlike functions. On varying the param- 
eters, he found the minimum of the energy curve (shown in 
Figure 43-4) to lie at t A b — 0.77 A, and to correspond to the 
value 4.00 v.e. for the dissociation energy D e of the molecule. 
This is an appreciable improvement, of 0.24 v.e., over the value 
given by Wang's function. The parameter values minimizing 

the energy 2 were found to be 
Z' = 1.193 and c = 0.256. 

It may be of interest to 
consider the hydrogen-mole- 
cule problem from another 
point of view. So far we have 
attempted to build a wave func- 
tion for the molecule from 
atomic orbital functions, a pro- 
cedure which is justified as a 
first approximation when r A B 
is large. This procedure, as 
generalized to complex mole- 
cules, is called the method of 



2.20 



w|. 



■2.24 



-2.28 



-2.32 



A 



1.00 



L75 



1.25 1.50 
r A67a -— 
Fig. 43-4. — Energy curves for the 
hydrogen molecule (in units e 2 /2ao) : A, 
for an extreme molecular-orbital wave 
function; B, for an extreme valence- 
bond wave function; and c, for a valence-bond wave functions, the 

valence-bond function with partial ionic _ rt _^ ™~^+,-_^ U^,*~~ „„~A i~ 
character ( Weinbaum) . name sometimes being used in 

the restricted sense of implying 
neglect of the ionic terms. Another way of considering the 
structure of complex molecules, called the method of molecular 
orbitals,* can be applied to the hydrogen molecule in the following 
way. Let us consider that for small values of K r AB the interaction 
of the two electrons with each other is small compared with their 
interaction with the two nuclei. Neglecting the term e 2 /r 12 
in the potential energy, the wave equation separates into equa- 
tions for each electron in the field of the two nuclei, as in the 
hydrogen-molecule-ion problem, and the unperturbed wave 
function for the normal state of the molecule is seen to be the 

1 S. Weinbaum, J. Chem. Phys. 1, 593 (1933). 

* Weinbaum also considered a more general function with different 
effective nuclear charges for the atomic and the ionic terms and found that 
this reduced to 43-13 on variation. 

« F. Hund, Z. f. Phys. 51, 759 (1928); 73, 1 (1931); etc.; R. S. Mulliken, 
Phys. Rev. 32, 186, 761 (1928); 41, 49 (1932); etc.; M. Dunkel, Z.f. phys. 
Chem. B7, 81; 10, 434 (1930); E. Huckel, Z.f. Phys. 60, 423 (1930); etc. 



XII-43b] THE HYDROGEN MOLECULE 347 

product of normal hydrogen-molecule-ion wave functions for the 
two electrons. Inasmuch as the function u u (1) + ^i« (1) 
is a good approximation to the wave function for the electron 
in the normal hydrogen molecule-ion, the molecular-orbital 
treatment corresponds to the wave function 

\uu A 0) + Uu M Q) i {uu A &) + u Ub {2) } (43-14) 

for the normal hydrogen molecule. It is seen that this is identical 
with Weinbaum's function 43-13 with c = 1; that is, with the 
ionic terms as important as the atomic terms. 

If the electric charges of the nuclei were very large, the inter- 
electronic interaction term would actually be a small perturba- 
tion, and the molecular-orbital wave function 43-14 would be a 
good approximation to the wave function for the normal state 
of the system. In the hydrogen molecule, however, the nuclear 
charges are no larger than the electronic charges, and the mutual 
repulsion of the two electrons may well be expected to tend to 
keep the electrons in the neighborhood of different nuclei, as 
in the simple Heitler-London-Wang treatment. It would be 
difficult to predict which of the two simple treatments is the 
better. On carrying out the calculations 1 for the molecular- 
orbital function 43-14, introducing an effective atomic number 
Z' y the potential curve A of Figure 43-4 is obtained, correspond- 
ing to r e = 0.73 A, D e = 3.47 v.e., and Z' = 1.193. It is seen 
that the extreme atomic-orbital treatment (the Wang curve) is 
considerably superior to the molecular-orbital treatment for the 
hydrogen molecule. 2 This is also shown by the results for the 
more general function 43-13 including ionic terms with a coeffi- 
cient c; the value of c which minimizes the energy is 0.256, 
which is closer to the atomic-orbital extreme (c = 0) than to the 
molecular-orbital extreme (c = 1). 

For the doubly charged helium molecule-ion, He^~+, a treatment 
Dased on the function 43-13 has been carried through, 3 leading 
to the energy curve shown in Figure 43-5. It is seen that at 
large distances the two normal He+ ions repel each other with the 
force e 2 /r 2 . At about 1.3 A the effect of the resonance integrals 

1 For this treatment we are indebted to Dr. S. Weinbaum. 

2 Similar conclusions are reached also when Z' is restricted to the value 
1 (Heitler-London treatment). 

*L. Pauling, /. Chem. Phys. 1, 56 (1933). 



348 



THE STRUCTURE OF SIMPLE MOLECULES [XII-43b 



becomes appreciable, leading to attraction of the two ions and a 
minimum in the energy curve at the predicted internuclear 
equilibrium distance r e = 0.75 A (which is very close to the value 
for the normal hydrogen molecule). At this distance the values 
of the parameters which minimize the energy are Z' = 2.124 
and c = 0.435. This increase in the value of c over that for the 
hydrogen molecule shows that as a result of the larger nuclear 
charges the ionic terms become more important than for the 
hydrogen molecule. 




Fig. 43-5. — The energy curve for normal He^+. 

We have discussed the extension of the extreme atomic-orbital 
treatment by the inclusion of ionic terms. A further extension 
could be made by adding terms corresponding to excited states 
of the hydrogen atoms. Similarly the molecular-orbital treat- 
ment could be extended by adding terms corresponding to 
excited states of the hydrogen molecule-ion. With these 
extensions the treatments ultimately become identical. 1 In 
the applications to complex molecules, however, it is usually 
practicable to carry through only the extremely simple atomic- 
orbital and molecular-orbital treatments; whether the slight 
superiority indicated by the above considerations for the atomic- 

1 See J. C. Slater, Phys. Rev. 41, 255 (1932). 



Xn-43c] 



THE HYDROGEN MOLECULE 



349 



orbital treatment is retained also for molecules containing 
atoms of larger atomic number remains an open question. 

So far we have not considered polarization of one atom by the 
other in setting up the variation function. An interesting 
attempt to do this was made by Rosen, 1 by replacing u u (1) in 
the Heitler-London-Wang function by u Uj (l) + (tu 2Pa (1) (with 
a similar change in the other functions), as in Dickinson's 
treatment of the hydrogen molecule-ion (which was suggested 
by Rosen's work). The effective nuclear charges Z'e in u u 
and Z"e in u 2p were assumed to be related, with Z" = 2Z'. 
It was found that this leads to an improvement of 0.26 v.e. in 
the value of D e over Wang's treatment, the minimum in the 
energy corresponding to the values r e = 0.77 A, D e = 4.02 v.e., 
Z' = 1.19, and a = 0.10. 

A still more general function, obtained by adding ionic terms 
(as in 43-13) to the Rosen function, was discussed by Weinbaum, 
who obtained D e = 4.10 v.e., Z' = 1.190, a- = 0.07, and c = 0.176. 

The results of the various calculations described in this section 
are collected in Table 43-1, together with the final results of 
James and Coolidge (see following section). 

Table 43-1. — Results of Approximate Treatments of the Normal 
Hydrogen Molecule 



Heitler-London-Sugiura 

Molecular-orbital treatment . . 

Wang 

Weinbaum (ionic) 

Rosen (polarization) 

Weinbaum (ionic-polarization) 

James-Coo lidge 

Experiment 



D e 



3. 14 v.e. 

3.47 

3.76 

4.00 

4.02 

4.10 

4.722 

4.72 



0.80 A 

0.73 

0.76 

0.77 

0.77 

0.74 
0.7395 



4800 era" 

4900 
4750 
4260 



4317.9 



1.193 

1.166 

1.193 

1.19 

1.190 



43c. The Treatment of James and Coolidge. — In none of the 
variation functions discussed in the preceding section does the 
intereleetronic interaction find suitable expression. A major 
advance in the treatment of the hydrogen molecule was made 
by James and Coolidge 2 by the explicit introduction of the 

1 N. Rosen, Phys. Rev. 38, 2099 (1931). 

a H. M. James and A. S. Coolidge, /. Chem. Phys. 1, 825 (1933). 



350 THE STRUCTURE OF SIMPLE MOLECULES [XII-43c 

interelectronic separation r i2 in the variation function (the 
similar step by Hylleraas having led to the ultimate solution 
of the problem of the normal helium atom). Using the elliptic 
coordinates (Sec. 42c) 

t r AX + r Bl r A2 + r B2 

Tab Tab 

Ta\ — r B \ Tai — r B 2 

rji = ; rj 2 = > 

Tab Tab 

and the new coordinate 

„ 2ri2 

U = y 

Tab 

James and Coolidge chose as the variation function the expression 

* = ^~ a(fl+f,) 2 Cmn; ^ (ff ^ r?Jl?7 ^ P + ^^"^ p ), (43-15) 

mnjkp 

the summation to include zero and positive values of the indices, 
with the restriction that j + k be even, which is required to make 
the function symmetric in the coordinates of the nuclei. 

Calculations were first made for r AB = 1.40a (the experi- 
mental value of r e ) and 8 = 0.75; with these fixed values the 
variation of the parameters can be carried out by the solution 
of a determinantal equation (Sec. 26d). It was found that 
five terms alone lead to an energy value much better than any 
that had been previously obtained, 1 the improvement being due 
in the main to the inclusion of one term involving u (Tables 
43-2 and 43-3). It is seen from Table 43-2 that the eleven-term 
and thirteen-term functions lead to only slightly different energy 
values, and the authors' estimate that the further terms will 
contribute only a small amount, making D e = 4.722 ± 0.013 v.e., 
seems not unreasonable. 

Using the eleven-term function, James and Coolidge investi- 
gated the effects of varying 5 and r AB , concluding that the 
values previously assumed minimize the energy, corresponding 
to agreement between the theoretical and the experimental 
value of r c , and that the energy depends on tab in such a way as to 
correspond closely to the experimental value of v e . 

1 It is of interest that the best value found by including only terms with 
p = is D e - 4.27 v.e., which is only slightly better than the best values 
of the preceding section. 



XII -43d] 



THE HYDROGEN MOLECULE 



351 



This must be considered as a thoroughly satisfactory treat- 
ment of the normal hydrogen molecule, the only improvement 
which we may look forward to being the increase in accuracy by 
the inclusion of further terms. 

Table 43-2. — Successive Approximations with the James-Coolidge 
Wave Function for the Hydrogen Molecule 



Number of terms 


Total energy 


D. 


1 

5 

11 

13 


-2.189 R H hc 
-2.33290 
-2.34609 
-2.34705 


2.56 v.e. 
4.504 
4.685 
4.698 



Table 43-3. — Values of Coefficients c mn jkp for Normalized Wave 
Functions for the Hydrogen Molecule* 



Term mnjkp 



00000 
00020 
10000 
00110 
00001 
10200 
10110 
10020 
20000 
00021 
10001 
00002 
00111 



Values of c mn jk P 



1 term 



1.69609 



5 terms 



2.23779 

0.80483 

-0.60985 

-0.27997 

0.19917 



11 terms 



2.29326 

1 . 19526 
-0.86693 
-0.49921 

0.33977 
-0.13656 

0.14330 
-0.07214 

0.06621 



-0.02456 
-0.03143 



13 terms 



2.22350 

1 . 19279 

-0.82767 

-0.45805 

0.35076 

-0.17134 

0.12394 

-0.12101 

0.08323 

0.07090 

-0.03987 

-0.01197 

-0.01143 



*In a later note, J. Chem. Phys. 3, 129 (1935), James and Coolidge state that these 
values are about 0.05 per cent too large. 

43d. Comparison with Experiment. — The theoretical values 
for the energy of dissociation of the hydrogen molecule and mole- 
cule-ion discussed in the preceding sections can be compared with 
experiment both directly and indirectly. The value 

Do = 2.639 ± 0.002 v.e. 
for H£ is in agreement with the approximate value 2.6 ±0.1 v.e. 
found from the extrapolated vibrational frequencies for excited 



352 



THE STRUCTURE OF SIMPLE MOLECULES [XII-43d 



states of H 2 . For the hydrogen molecule the energy calculations 
of James and Coolidge with an estimate of the effect of further 
terms and corrections for zero-point vibration (using a Morse 
function) and for the rapid motion of the nuclei (corresponding 
to the introduction of a reduced mass of electron and proton) 
lead to the value 1 4.454 ± 0.013 v.e. for the dissociation energy 
O . This is in entire agreement with the most accurate experi- 



Ht + E" 



H+H + +E" 














I 


v / 


i 




D e (H 
v=o 


2 + ) 


) 


Do(H e 


e(H 2 + ) 




i 


i ' 


A 






4hi 








1(H) 




KH 2 ) 








H+H 






1 


i 






D e (H 2 ) 


[ 


w 


) 


' 


u 




H» 


' ,'-t 



^hT) e (H 2 ) 

Fig. 43-6. — Energy-level diagram for a system of two electrons and two protons. 



mental value, 4.454 ± 0.005 v.e., obtained by Beutler 2 by the 
extrapolation of observed vibrational levels. 

Another test of the values can be made in the following way. 
From the energy-level diagram for a system of two electrons and 



1 H. M. James and A. S. Coolidge, J. Chem. Phys. 3, 129 (1935). We are 
indebted to Drs. James and Coolidge for the personal communication of this 
and other results of their work. 

2 H. Beutler, Z. phys. Chem. B27, 287 (1934). A direct thermochemical 
determination by F. R. Bichowsky and L. C. Copeland, Jour. Am. Chem. 
Soc. 50. 1315 (1928), gave the value 4.55 ± 0.15 v.e. 



XII-43e] THE HYDROGEN MOLECULE 353 

two protons shown in Figure 43-6 we see that the relation 

J(H 2 ) + D (H+) = /(H) + D (H 2 ) (43-16) 

holds between the various ionization energies and dissociation 
energies. With the use of the known values of 7(H) and D (HJ) 
(the latter being the theoretical value) and of the extrapolated 
spectroscopic value of 7(H 2 ), £>o(H 2 ) is determined 1 as 

4.448 ± 0.005 v.e., 

again in excellent agreement with the value given by James 
and Coolidge. 

43e. Excited States of the Hydrogen Molecule. — Several 
excited states of the hydrogen molecule have been treated by 
perturbation and variation methods, 2 with results in approximate 
agreement with experiment. 

Instead of discussing these results, let us consider the simple 
question as to what wave functions for the hydrogen molecule 
can be built from Is hydrogenlike functions u, L and u B alone. 
There are four product functions of this type, u A (l)u B (2), 
u B (l)u A (2), u A (l)u A (2), and u B (\)u B (2). The equivalence of 
the two electrons and of the two nuclei requires that the wave 
functions obtained from these by solution of the secular equation 
be either symmetric or antisymmetric in the positional coordi- 
nates of the two electrons and also either symmetric or antisym- 
metric in the two nuclei. These functions are 



n}' 



^(1)^(2) + u B (l)u A (2)\, {u a (\)u a (2) +^(1)^(2)) 



\S N S E1 2+, 

III ^(1)^(2) - u B (l)u A (2),A N A E *Xt, 

IV u A (l)u A (2) - u B (l)u B (2),A»S* l 2i 9 

functions I and II being formed by linear combination of the two 
indicated functions. One of these (I, say) represents the 

1 Personal communication to Dr. James from Prof. O. W. Richardson. 

2 E. C. Kemble and C. Zener, Phys. Rev. 33, 512 (1929); C. Zener and 
V\ Guillemin, Phys. Rev. 34, 999 (1929); E. A. Hylleraas, Z.f. Phys. 71,- 
739 (1931); E. Majorana, Atti Accad. Lincei 13, 58 (1931); J. K. L. Mac 
Donald, Proc. Roy. Soc. A136, 528 (1932). The method of James and 
Coolidge has been applied to several excited states of the hydrogen molecule 
by R. D. Present, J. Chem. Phys. 3, 122 (1935), and by H. M. James, A. S. 
Coolidge, and R. D. Present, in a paper to be published soon. 



354 THE STRUCTURE OF SIMPLE MOLECULES [XH-486 

normal state of the molecule (Sec. 436, Weinbaum), and the other 
an excited state. The term symbol ^^ for these states contains 
the letter 2 to show that there is no component of electronic 
orbital angular momentum along the nuclear axis; the superscript 

I to show that the molecule is in a singlet state, as shown also 
by the symbol S B , meaning symmetric in the positional coordi- 
nates of the two electrons, Pauli's principle then requiring that 
the electron-spin function b6 the singlet function 

a(l)0(2) -/9(l)a(2); 

and the superscript + to show that the electronic wave function 
is symmetric in the two nuclei, as shown also by S N . In 
addition the subscript g (German gerade) is given to show that 
the electronic wave function is an even function of the electronic 
coordinates. Functions III and IV are both antisymmetric in 
the nuclei, as indicated by the symbol A N and the superscript — , 
and are odd functions, as shown by the subscript u (German 
ungerade), III being a triplet and IV a singlet function. A 
further discussion of these symmetry properties will be given in 
the next section and in Section 48. 

Function III represents the repulsive interaction of two 
normal hydrogen atoms, as mentioned in Section 43a. Function 

II is mainly ionic in character and function IV completely so, 
representing the interaction of H+ and H~\ Of these IV cor- 
responds to a known state, the first electronically excited state 
of the molecule. As might have been anticipated from the 
ionic character of the wave function, the state differs in its prop- 
erties from the other known excited states, having r, = 1.29 A 
and v = 1358 cm -1 , whereas the others have values of r e and v e 
close to those for the normal hydrogen molecule-ion, 1.06 A 
and 2250 cm - " 1 . The calculations of Zener and Guillemin and 
of Hylleraas have shown that at the equilibrium distance the 
wave function for this state involves some contribution from 
wave functions for one normal and one excited atom (with 
n = 2, 1 = 1), and with increase in r A B this contribution increases, 
the molecule in this state dissociating into a normal and an 
excited atom. 

The state corresponding to II has not yet been identified. 

Problem 48-1. Construct a wave function of symmetry type A N S* 
from 1* and 2p functions. 



Xn-43f] THE HYDROGEN MOLECULE 355 

43f. Oscillation and Rotation of the Molecule. Ortho and 
Para Hydrogen. — In accordance with the discussion of the pre- 
ceding sections and of Chapter X, we can represent the complete 
wave function for the hydrogen molecule as the product of 
five functions, one describing the orbital motion of the electrons, 
the second the orientation of electron spins, the third the oscilla- 
tional motion of the nuclei, the fourth the rotational motion of 
the nuclei, and the fifth the orientation of nuclear spins (assuming 
them to exist) : 

(electronic\ / electronic- \ , , w . ./ nuclear- \ 
, . , y Y nuclear \/ nuclear \# \ 

A . \ .. jl oscillation I l rotation )\ . . .. } 
motion /\onentation/\ /\ / ^orientation / 

For the normal electronic state the first of these is symmetric 
in the two electrons, the second antisymmetric, and the remaining 
three independent of the electrons (and hence symmetric), 
making the entire function antisymmetric in the two electrons, 
as required by Pauli's principle. Let us now consider the sym- 
metry character of these functions with respect to the nuclei. 
The first we have seen to be symmetric in the nuclei. The 
second is also symmetric, not being dependent on the nuclear 
coordinates. The third is also symmetric for all oscillational 
states, inasmuch as the variable r which occurs in the oscilla- 
tional wave function is unchanged by interchanging the nuclei. 
The rotational function, however, may be either symmetric or 
antisymmetric. Interchanging the two nuclei converts the 
polar angle # into t — # and <p into w + <P) the consideration of 
the rotational wave functions (Sees. 35a and 21) shows that 
this causes a change in sign if the rotational quantum number K 
is odd, and leaves the function unchanged if K is even. Hence 
the rotational wave function is symmetric in the nuclei for even 
rotational states and antisymmetric for odd rotational states. 
The nuclear-spin function can be either symmetric or antisym- 
metric, providing that the nuclei possess spins. 

•By an argument identical with that given in Section 296 for 
the electrons in the helium atom we know that a system con- 
taining two identical protons can be represented either by wave 
functions which are symmetric in the protons or by wave func- 
tions which are antisymmetric in the protons. Let us assume 



356 THE STRUCTURE OF SIMPLE MOLECULES [XII-43f 

that the protons possessed no spins and that the symmetric 
functions existed in nature. Then only the even rotational 
states of the normal hydrogen molecule would occur (and only 
the odd rotational states of the A N electronic state IV of the 
preceding section). Similarly, if the antisymmetric functions 
existed in nature, only the odd rotational states of the normal 
molecule would occur. If, on the other hand, the protons 
possessed spins of % (this being the value of the nuclear-spin 
quantum number /), both even and odd rotational states would 
occur, in the ratio of 3 to 1 if the complete wave function were 
symmetric or 1 to 3 if it were antisymmetric, inasmuch as there 
are for I = }i three symmetric nuclear-spin wave functions, 

a(A) a(B), 

-±={a(A)P(B)+P(A)a{B)}, 

and 

KA) KB), 
and one antisymmetric one, 

-^={«(A) |8(B) -j8(A)a(fl)}. 

In this case, then, we would observe alternating intensities in 
the rotational fine structure of the hydrogen bands, with the 
ratio of intensities 3:1 or 1:3, depending on the symmetry char- 
acter of protons. Similar alternating intensities result from 
larger values of /, the ratio being 1 / + 1 to /. It is seen that 

1 Thus for / = 1 there are three spin functions for one particle, a, /3, and 7, 
say, corresponding to m/ = -f 1, 0, — 1. From these we can build the 
following wave functions for two particles, giving the ratio 2:1. 

Symmetric Antisymmetric 

*(A) a(B) 
KA) fi(B) 
7(A) 7(B) 

'a(A) ^(B) + fi(A) a(B)} ~^=\a(A) /3(B) - /3(A) a(B)} 



s/2 \/2 

■4=|a(A) 7(B) + 7(A) a(B)\ ~ 

V2 V2 

-^{0(A)7(B) + 7(A) /3(B)} -U 

V2 V2 



\a(A) 7(B) + 7(A) a(B)\ -W«(A) 7(B) - 7(A) «(B) ) 

{/3(A) 7(B) 4- 7(A) /3(B)} -^=|0(A) 7(B) - 7(A) 0(B)] 



XII-43f] THE HYDROGEN MOLECULE 357 

from the observation and analysis of band spectra of molecules 
containing two identical nuclei the symmetry character and the 
spin of the nuclei can be determined. 

It was found by Dennison 1 (by a different method — the study 
of the heat capacity of the gas, discussed in Section 49e) that 
protons (like electrons) have a spin of one-half, and that the 
allowed wave functions are completely antisymmetric in the 
proton coordinates (positional plus spin). This last statement 
is the exact analogue of the Pauli exclusion principle. 2 

Each of the even rotational wave functions for the normal 
hydrogen molecule is required by this exclusion principle to be 
combined with the antisymmetric spin function, whereas each 
of the odd rotational wave functions can be associated with the 
three symmetric spin functions, giving three complete wave 
functions. Hence on the average there are three times as many 
complete wave functions for odd rotational states as for even, and 
at high temperatures three times as many molecules will be in 
odd as in even rotational states (Sec. 49e). Moreover, a molecule 
in an odd rotational state will undergo a transition to an even 
rotational state (of the normal molecule) only extremely rarely, 
for such a transition would result only from a perturbation involv- 
ing the nuclear spins, and these are extremely small in magnitude. 
Hence (as was assumed by Dennison) under ordinary circum- 
stances we can consider hydrogen as consisting of two distinct 
molecular species, one, called para hydrogen, having the nuclear 
spins opposed and existing only in even rotational states (for 
the normal electronic state), and the other, called ortho hydrogen, 
having the nuclear spins parallel and existing only in the odd 
rotational states. Ordinary hydrogen is one-quarter para and 
three-quarters ortho hydrogen. 

On cooling to liquid-air temperatures the molecules of para 
hydrogen in the main go over to the state with K = and 
those of ortho hydrogen to the state with K — 1, despite the 
fact that at thermodynamic equilibrium almost all molecules 
would be in the state with K = 0, this metastable condition 
being retained for months. It was discovered by Bonhoeffer 

1 D. M. Dennison, Proc. Roy. Soc. A115, 483 (1927). 

2 The spins and symmetry nature for other nuclei must at present be 
determined experimentally; for example, it is known that the deuteron has 
/ = 1 and symmetric wave functions. 



358 THE STRUCTURE OF SIMPLE MOLECULES [Xn-44a 

and Harteck, 1 however, that a catalyst such as charcoal causes 
thermodynamic equilibrium to be quickly reached, permitting 
the preparation of nearly pure para hydrogen. It is believed 
that under these conditions the ortho-para conversion is due to 
a magnetic interaction with the nuclear spins, 2 and not to dis- 
sociation into atoms and subsequent recombination, inasmuch as 
the reaction H2 + D 2 ^ 2HD is not catalyzed under the same 
conditions. The conversion is catalyzed by paramagnetic sub- 
stances 3 (oxygen, nitric oxide, paramagnetic ions in solution), and 
a theoretical discussion of the phenomenon has been published. 4 
At higher temperatures the conversion over solid catalysts seems 
to take place through dissociation and recombination. 

44. THE HELIUM MOLECULE-ION Hef AND THE INTERACTION 
OF TWO NORMAL HELIUM ATOMS 

In the preceding sections we have discussed systems of two 
nuclei and one or two electrons. Systems of two nuclei and three 
or four electrons, represented by the helium molecule-ion He£ 
and by two interacting helium atoms, respectively, are treated 
in the following paragraphs. A discussion of the results obtained 
for systems of these four types and of their general significance 
in regard to the nature of the chemical bond and to the structure 
of molecules will then be presented in Section 45. 

44a. The Helium Molecule -ion He£. — In treating the system 
of two helium nuclei and three electrons by the variation method 
let us first construct electronic wave functions by using only 
hydrogenlike Is orbital wave functions for the two atoms, which 
we may designate as uj. and u By omitting the subscripts Is for 
the sake of simplicity. Four completely antisymmetric wave 
functions can be built from these and the spin functions a and f$. 
These are (before normalization) 



ft- 



Ma(1)«(1) u A (l)p(l) u B (l)a(l) 
u A (2)a(2) 11,(2)0(2) u B {2) «(2) 
u A (Z) a(3) u A (Z) 0(3) ti B (3) a(3) 



(44-1) 



1 K. F. Bonhoeffbr and P. Harteck, Z.f. phys. Chem. B4, 113 (1929). 
* K. F. Bonhoeffer, A. Farkas, and K. W. Rummbl, Z. f. phys. Chem. 
B21, 225 (1933). 

«L. Farkas and H. Sachssb, Z. /. phys. Chem. B23, 1, 19 (1933). 
« E. Wigner, Z. /. phys. Chem. B23, 28 (1933). 



XII-44a] THE HELIUM MOLECULE-ION 359 

and 



^ii 



u B (l)a(l) u M (l)fi(l) u A (l)«(l) 
iij(2)a(2) Ub(2)j8(2) u A (2)a(2) 
u B (%) a(3) 1*^(3) 0(3) iu(3) a(3) 



(44-2) 



and two other functions, ^m and ^iv, obtained by replacing 
a by in the last column of these functions. It is seen that the 
function ^i represents a pair of electrons with opposed spins on 
nucleus A (as in the normal helium atom) and a single electron 
with positive spin on nucleus B; this we might write as He: 
•He + . Function ^n similarly represents the structure He + :He, 
the nuclei having interchanged their roles. It is evident that 
this system shows the same degeneracy as the hydrogen molecule- 
ion, and that the solution of the secular equation for \f/i and ^n 
will lead to the functions yf/ s and ^a, the nuclear-symmetric and 
nuclear-antisymmetric combinations of ^i and ^n (their sum 
and difference), as the best wave functions given by this approxi- 
mate treatment. The other wave functions ^ m and ^iv lead 
to the same energy levels. 

The results of the energy calculation 1 (which, because of its 
similarity to those of the preceding sections, does not need to be 
given in detail) are shown in Figure 44-1. It is seen that the 
nuclear-antisymmetric wave function $ A corresponds to repulsion 
at all distances, whereas the nuclear-symmetric function ^ s 
leads to attraction and the formation of a stable molecule-ion. 
That this attraction is due to resorance between the structures 
He: He + and He- + :He is shown by comparison with the energy 
curve for fa or ^n alone, given by the dashed line in Figure 44-1. 
We might express this fact by writing for the normal helium 
molecule-ion the structure He—He*, and saying that its stabil- 
ity is due to the presence of a three-electron bond between the 
two atoms. 

The function yps composed of Is hydrogenlike orbital wave 
functions with effective nuclear charge 2e leads to a minimum 
in the energy curve at r e = 1.01 A and the value 2.9 v.e. for the 
energy of dissociation D e into He + He 4 *. A more accurate 
treatment 2 can be made by minimizing the energy for each value 

l L. Paulino, /. Chem. Phys. 1, 56 (1933). 

1 L. Pauling, loc. cit. The same calculation with Z' given the fixed value 
1.8 was made by E. Majorana, Nuovo Cim. 8, 22 (1931). 



360 



THE STRUCTURE OF SIMPLE MOLECULES [XII-44a 



of Tab with respect to an effective nuclear charge Z'e. This 
leads to r e = 1.085 A, D e = 2.47 v.e., and the vibrational 
frequency v e = 1950 cm -1 , with Z' equal to 1.833 at the equilib- 
rium distance. A still more reliable treatment can be made 
by introducing two effective nuclear charges Z'e and Z"e, one for 
the helium atom and one for the ion, and minimizing the energy 
with respect to Z f and Z". This has been done by Weinbaum, 1 



10 






-5 





I T 
1 \ 

I \ 
\ \ 
\ \ 
\ \ 
\ \ 
\ \ 








\ \ 
\ \ 

\ N 
V \ 


\a n 






X. S 


1^^^^ 













05 



1.0 
r-* 
AB 



15 



2.0A 



Fig. 44-1. — Energy curves for attractive and repulsive states of Het- The 
dashed curve corresponds to a non-existent state, resonance between He: -He + 
and He« + :He being excluded. 



who obtained the values r e = 1.097 A, D e = 2.22 v.e., Z' = 1.734, 
and Z" = 2.029. The results of these calculations are in good 
agreement with the experimental values given by excited states 
of the diatomic helium molecule (consisting of the normal mole- 
cule-ion and an outer electron), which are r e = 1.09 A, D e = 2.5 
v.e., and v e = 1650 cm -1 . 

It is of interest that the system of a helium nucleus and 
a hydrogen nucleus and three electrons does not show the 
degeneracy of functions fa and ^ n , and that in consequence the 
interaction of a normal helium atom and a normal hydrogen 

i S. Weinbaum, /. Chem. Phys. 3, 547 (1935). 



XII-44b] THE HELIUM MOLECULE-ION 361 

atom corresponds to repulsion, as has been verified by approxi- 
mate calculations. 1 
44b. The Interaction of Two Normal Helium Atoms.— We may 

write for the wave function for the normal state of a system 
consisting of two nuclei and four electrons the expression 



+ -N 



u*(l)«(l) «a(1)j8(1) m*(1)«(1) «,(l)j8(l) 

u A (2)a(2) «a(2)i8(2) u B (2)a{2) u B {2) 0(2) 

u A (3)a(3) ujl(S)P(Z) u B (3)a(3) u*(3) 0(3) 

u A (l)a(4) 1^(4)0(4) «*(4) a(4) w B (4) j8(4) 



, (44-3) 



in which u A and u B represent Is wave functions about nuclei 
A and B, respectively, and iV is a normalizing factor. This wave 
function satisfies Pauli's principle, being completely anti- 
symmetric in the four electrons. It is the only wave function 
of this type which can be constructed with the use of the one- 
electron orbital functions u A and u B alone. 

It was mentioned by Heitler and London in their first paper 1 
that rough theoretical considerations show that two normal 
helium atoms repel each other at all distances. The evaluation 
of the energy for the wave function \p of Equation 44-3 with 
iia and u B hydrogenlike Is wave functions with effective atomic 
number Z' = 2 %6 was carried out by Gentile. 2 A more 
accurate calculation based on a helium-atom wave function not 
given by a single algebraic expression has been made by Slater, 3 
who found that the interaction energy is given by the approxi- 
mate expression 

2.43ft 

W - W ° = 7.70 • 10- 10 e ao ergs. (44-4) 

This represents the repulsion which prevents the helium atoms 
from approaching one another very closely. The weak attrac- 
tive forces which give rise to the constant a of the van der Waals 
equation of state cannot be treated by a calculation of this type 
based on unperturbed helium-atom wave functions. It will 
be shown in Section 476 that the van der Waals attraction is 

given approximately by the energy term — 1.41e 2 ^or — 0.607p| 

1 W. Heitler and F. London, Z. f. Phys. 44, 455 (1927). 

2 G. Gentile, Z. f. Phys. 63, 795 (1930). 
8 J. C. Slater, Phys. Rev. 32, 349 (1928). 



362 THE STRUCTURE OF SIMPLE MOLECULES [XII-45 

ergs. The equilibrium interatomic distance corresponding to 
this attraction term and the repulsion term of Equation 44-4 
is 3.0 A, in rough agreement with the experimental value of 
about 3.5 A for solid helium, showing that the theoretical calcu- 
lations are of the correct order of magnitude. 

45. THE ONB-BLBCTRON BOND, THE ELECTRON-PAIR BOND, 
AND THE THREE-ELECTRON BOND 

In the preceding sections we have discussed systems containing 
two nuclei, each with one stable orbital wave function (a Is 
function), and one, two, three, or four electrons. We have found 
that in each case an antisymmetric variation function of the 
determinantal type constructed from atomic orbitals and 
spin functions leads to repulsion rather than to attraction 
and the formation of a stable molecule. For the four-electron 
system only one such wave function can be constructed, so that 
two normal helium atoms, with completed K shells, interact with 
one another in this way. For the other systems, on the other 
hand, more than one function of this type can be set up (the two 
corresponding to the structures H- H+ and H + H for the hydro- 
gen molecule-ion, for example); and it is found on solution 
of the secular equation that the correct approximate wave 
functions are the sum and difference of these, and that in each 
case one of the corresponding energy curves leads to attraction 
of the atoms and the formation of a stable bond. We call 
the bonds involving two orbitals (one for each nucleus) and one, 
two, and three electrons the one-electron bond, the electron-pair 
bond, and the three-electron bond, respectively. 

The calculations for the hydrogen molecule-ion, the hydrogen 
molecule, and the helium molecule-ion show that for these 
systems the electron-pair bond is about twice as strong a bond 
(using the dissociation energy as a measure of the strength of a 
bond) as the one-electron bond or the three-electron bond. 1 This 
fact alone provides us with some explanation of the great impor- 
tance of the electron-pair bond in molecular structure in general 
and the subsidiary roles played by the one-electron bond and the 
three-electron bond. 2 

1 See, however, the treatment of Li"^ by H. M. James, /. Chem. Phys. 3, 
9 (1936). 

*L. Pauling, /. Am. Chem. Soc. 53, 3225 (1931). 



XII-46] THE ONE-ELECTRON BOND 363 

There is a still more cogent reason for the importance of the 
electron-pair bond. This is the nature of the dependence of the 
energy of the bond on the similarity or dissimilarity of the two 
nuclei (or the two orbitals) involved. Using only two orbitals, 
u A and u B , we can construct for the one-electron system only the 
two wave functions 

fa = u A (l) «(1) 
and 

fa = u B (l) «(1) 

(together with two others involving 0(1) which do not combine 
with these and which lead to the same energy curves). These 
correspond to the electronic structures A- B+ and A + \B. If 
A and B are identical (or if fa and fai correspond to the same 
energy because of an accidental relation between the orbitals 
and the nuclear charges) there is degeneracy, and the interaction 
of fa and fa causes the formation of a stable one-electron bond. 
If this equality of the energy does not obtain, the bond is weak- 
ened, the bond energy falling to zero as the energy difference for 
\pi and 1^11 becomes very large. 

The three-electron bond behaves similarly. The wave func- 
tions (Eqs. 44-1 and 44-2) are closely related to those for the 
one-electron system, and the bond energy similarly decreases 
rapidly in magnitude as the energy difference for the two wave 
functions increases. Hence, in general, we expect strong one- 
electron bonds and three-electron bonds not to be formed 
between unlike atoms. 

The behavior of the electron-pair bond is entirely different. 
The principal degeneracy leading to bond formation is that 
between the wave functions 



and 

fa = 



Ml)«(l) Ub(1)j8(1) 

14,(2) a(2) v«(2)j8(2) 

tfc(l)j8(l) t*B(l)a(l) 

u A (2)fi(2) ti B (2)a(2) 



These correspond to the same energy value even when A and B 
are not identical; hence there is just the same resonance stabiliz- 
ing an electron-pair bond between unlike atoms as between like 
atoms. Moreover, the influence of the ionic terms is such as to 



364 THE STRUCTURE OF SIMPLE MOLECULES [XII-45 

introduce still greater stability as the nuclei become more unlike. 
One of the ionic wave functions 



*m = u A (l) u a (2) 
and 

tiv = u B (l) u b (2) 



«(1) 18(1) 
a(2) 0(2) 



'«(1)18(1) 
«(2) /8(2) 

corresponding to the ionic structures A : ~ B+ and A + : B~~, becomes 
more and more important (contributing more and more to the 
normal state of the molecule) as one of the atoms becomes more 
electronegative than the other, in consequence of the lowering 
of the energy for that ionic function. This phenomenon causes 
electron-pair bonds between unlike atoms to be, in general, 
somewhat stronger than those between like atoms. The dis- 
cussion of this subject has been in the main empirical. 1 

It has been found possible to apply quantum-mechanical 
methods such as those described in this chapter in the detailed 
discussion of the electronic structure of polyatomic molecules 
and of valence and chemical bond formation in general. Only 
in a very few cases has the numerical treatment of polyatomic 
molecules been carried through with much accuracy; the most 
satisfactory calculation of this type which has been made is 
that of Coolidge 2 for the water molecule. General arguments 
have been presented 3 which provide a sound formal justification 
for the postulates previously made by the chemist regarding the 
nature of valence. It can be shown, for example, that one bond 
of the types discussed in this section can be formed by an atom 
for each stable orbital of the atom. Thus the first-row elements 
of the periodic system can form as many as four bonds, by using 
the four orbitals of the L shell, but not more. This result and 
other results 4 regarding the relative orientation of the bond 
axes provide the quantum-mechanical basis for the conception 
of the tetrahedral carbon atom. Special methods for the 

l L. Pauling, /. Am. Chem. Soc. 54, 3570 (1932). 
2 A. S. Coolidge, Phys. Rev. 42, 189 (1932). 

* W. Heitler, Z. f. Phys. 47, 835 (1928), etc.; F. London, Z. f. Phys. 
60, 24 (1928), etc.; M. Born, Z.f. Phys. 64, 729 (1930); J. C. Slater, Phys. 
Rev. 38, 1109 (1931). 

* J. C. Slater. Phys. Rev. 34, 1293 (1929); L. Pauling, /. Am. Chem. Soc, 
53, 1367 (1931); J. H. Van Vleck, /. Chem. Phys. 1, 177 (1933), etc. 



XII-45] THE ONE-ELECTRON BOND 365 

approximate treatment of the stability of very complex molecules 
such as the aromatic hydrocarbons 1 have also been developed 
and found to be useful in the discussion of the properties of these 
substances. The already very extensive application of wave 
mechanics to these problems cannot be adequately discussed 
in the small space which could be allowed it in this volume. 

1 E. Huckel, Z. f. Phys. 70, 204 (1931), etc.; G. Rumer, Gottinger Nachr. 
p. 337, 1932; L. Pauling, J. Chem. Phys. 1, 280 (1933); L. Pauling and 
G. W. Wheland, ibid. 1, 362 (1933); L. Pauling and J. Sherman, ibid. 1, 
679 (1933), etc. 



CHAPTER XIII 

THE STRUCTURE OF COMPLEX MOLECULES 

In carrying out the simple treatments of the hydrogen mole- 
cule-ion, the hydrogen molecule, the helium molecule-ion, 
and the system composed of two normal helium atoms discussed 
in the last chapter, we encountered no difficulty in constructing a 
small number of properly antisymmetric approximate wave 
functions out of one-electron orbital functions for the atoms of 
the molecule. The same procedure can be followed for more 
complex molecules; it is found, however, that it becomes so 
complicated as to be impracticable for any but the simplest 
molecules, unless some method of simplifying and systematizing 
the treatment is used. A treatment of this type, devised by 
Slater, 1 is described in the following sections, in conjunction 
with the discussion of a special application (to the system of 
three hydrogen atoms). Slater's treatment of complex mole- 
cules has been the basis of most of the theoretical work which has 
been carried on in this field in the last three years. 

46. SLATER'S TREATMENT Ot COMPLEX MOLECULES 

In the last chapter we have seen that a good approximation 
to the wave function for a system of atoms at a considerable 
distance from one another is obtained by using single-electron 
orbital functions u a (l), etc., belonging to the individual atoms, 
and combining them with the electron-spin functions a and p 
in the form of a determinant such as that of Equation 44-3. 
Such a function is antisymmetric in the electrons, as required by 
Pauli's principle, and would be an exact solution of the wave 
equation for the system if the interactions between the electrons 
and those between the electrons of one atom and the nuclei 
of the other atoms could be neglected. Such determinantal 

1 J. C. Si-atbb. Phya. Rev. 38, 1109 (1931). 

366 



XIII-46] SLATER'S TREATMENT OF COMPLEX MOLECULES 367 

functions are exactly analogous to the functions 1 used in Section 
30a in the treatment of the electronic structure of atoms. 

It may be possible to construct for a complex molecule many 
such functions with nearly the same energy, all of which would 
have to be considered in any satisfactory approximate treatment. 
Thus if we consider one atom to have the configuration ls 2 2« 2 2p, 
we must consider the determinantal functions involving all three 
2p functions for that atom. A system of this type, in which 
there are a large number of available orbitals, is said to involve 
orbital degeneracy. Even in the absence of orbital degeneracy, 
the number of determinantal functions to be considered may be 
large because of the variety of ways in which the spin functions 
a and can be associated with the orbital functions. This 
spin degeneracy has been encountered in the last chapter; in the 
simple treatment of the hydrogen molecule we considered the 
two functions corresponding to associating positive spin with 
the orbital u A and negative spin with u B , and then negative spin 
with u A and positive spin with u B (Sec. 45). The four wave 
functions described in Section 44a for the helium molecule-ion 
might be represented by the scheme of Table 46-1. The plus 

Table 46-1. — Wave Functions for the Helium Moleculb-ion, .HbJ" 



Function 


u A 


UB 


2m. 


I 


+ - 


+ 


+K 


II 


+ 


+ - 


+H 


III 


+ - 


— 


-H 


IV 


— 


+ - 


-X 



and minus signs show which spin function a or is to be asso- 
ciated with the orbital functions u A and u B (in this case Is func- 
tions on the atoms A and B, respectively) in building up the 
determinantal wave functions. Thus row 1 of Table 46-1 
corresponds to the function fa given in Equation 44-1. 

The column labeled Sm a has the same meaning as in the atomic 
problem; namely, it is the sum of the z-components of the spin 
angular momentum of the electrons (with the factor h/2nr) m 
Just as in the atomic case, the wave functions which have different 

1 In Section 30a the convention was adopted that the symbol Ua(i) should 
include the spin function a(i) or 0(i). In this section we shall not use the 
convention, instead writing the spin function a or explicitly each time. 



368 



THE STRUCTURE OF COMPLEX MOLECULES [XIII-46a 



values of 2ra, do not combine with one another, so that we were 
justified in Section 44a in considering only ^ r and \p n . 

Problem 4G-1. Set up tables similar to Table 46-1 for the hydrogen 
molecule using the following choices of orbital functions: (a) Is orbitals 
on the two atoms, allowing only one electron in each, (b) The same 
orbitals but allowing two electrons to occur in a single orbital also; i.e., 
allowing ionic functions, (c) The same as (a) with the addition of func- 
tions 2p, on each atom, (d) The molecular orbital (call it u) obtained by 
the accurate treatment of the normal state of the hydrogen molecule-ion. 

46a. Approximate Wave Functions for the System of Three 
Hydrogen Atoms. — In the case of three hydrogen atoms we can 
set up a similar table, restricting ourselves to the three Is func- 
tions w a , u b} and u c on three atoms a, b, and c, respectively, and 
neglecting ionic structures (Table 46-2). 

Table 46-2. — Wave Functions for the System of Three Hydrogen 

Atoms 



Function 


U a 


Ub 


Uc 


Sm, 


I 


+ 


+ 


+ 


+ « 


II 


+ 


+ 


- 


4-3^ 


III 


+ 


- 


+ 


+H 


IV 


- 


+ 


+ 


■VYi 


V 


+ 


~ 


- 


- l A 


VI 


- 


+ 


- 


~K 


VII 


— 


- 


+ 


-H 


VIII 


— 


— 


— 


-H 



The wave function corresponding to row II of Table 46-2 is, for 
illustration, 



^ii = 



1 



VsH 



U a (l)a(l) u b (l)a(l) u e (l)0(l) 
u (2) a(2) u h (2) a(2) u c (2) fi(2) 
u«(3) a(3) u b (3) a(3) u e (3) 0(3) 



(46-1) 



Each of the functions described in Table 46-2 is an approxi- 
mate solution of the wave equation for three hydrogen atoms; 
it is therefore reasonable to consider the sum of them with 
undetermined coefficients as a linear variation function. The 
determination of the coefficients and the energy values then 
requires the solution of a secular equation (Sec. 26d) of eight 
rows and columns, a typical element of which is 

#m - AinW (46-2) 



XHI-46b] SLATER'S TREATMENT OF COMPLEX MOLECULES 369 

where 

#m « JWiMr, (46-3) 

and 

Ai ir = jWudr, (46-4) 

H being the complete Hamiltonian operator for the system. 

Problem 46-2. Make a table similar to Table 46-2 but including all 
ionic functions that can be made with the use of u a , Ub, and u e . 

46b. Factoring the Secular Equation. — In the discussion of the 
electronic structure of atoms (Sec. 30c) we found that the 
secular equation could be factored to a considerable extent 
because integrals involving wave functions having different 
values of Sm, or different values of 2m z (the .quantum numbers 
of the components of spin and orbital angular momentum, 
respectively) are zero. In the molecular case the orbital angular 
momentum component is no longer a constant of the motion 
(Sec. 52), so that only the spin quantum numbers are useful 
in factoring the secular equation. 

In the case of the system under discussion, we see from Table 
46-2 that the secular equation factors into two linear factors 
(2ra, = % and —%) and two cubic factors (2w, = Yi 
and — J^). On the basis of exactly the Same reasoning as used 
in Section 30c for the atomic case, we conclude that the roots of 
the two linear factors will be equal to each other and also to 
one of the roots of each of the cubic factors. 1 The four cor- 
responding wave functions are therefore associated with a quartet 
energy level, which on the vector picture corresponds to the 
parallel orientation of the three spin vectors, the four states 
differing only in the orientation of the resultant vector. 

The two remaining energy levels will occur twice, once in 
each of the cubic factors. Each of them is, therefore, a doublet 
level. The straightforward way of obtaining their energy values 
would be to solve the cubic equation; but this is unnecessary, 
inasmuch as by taking the right linear combinations of II, III, 
and IV it is possible to factor the cubic equation into a linear 
factor and a quadratic factor, the linear factor yielding the 
energy of the quartet level. Such combinations are 

1 These statements can easily be verified by direct comparison of the 
roots obtained, using the expressions for the integrals given in Section 46c. 



370 



THE STRUCTURE Of COMPLEX MOLECULES [XHI-46c 



.1 = 



vf 



(II - III), 



B 



= -t=(III 

V2 



IV), 



and 



c - -^(iv - id, 

Z> = -L(II + III + IV). 



(46-5) 
(46-6) 



(46-7) 



(46-8) 



Since these four functions are constructed from only three linearly- 
independent functions II, III, and IV, they cannot be linearly 
independent; in fact, it is seen that A + B + C = 0. The 
factoring of the secular equation will be found to occur when it 
is set up in terms of D and any two of the functions A, B f and C. 
The energy of the quartet level can be obtained from either of 
the linear factors; it is given by the relation 



An 



(46-9) 



The values of the energy of the two doublet levels are obtained 
from the quadratic equation 

H AA - AaaW Hab - AabW 
Hba ~ &baW Ebb - &bbW 



o, 



(46-10) 



in which 



Hab = JA*HBdT,) 

Aab - JA*Bdr, > 



(46-11) 



Problem 46-3. Indicate how the secular equation for each of the cases 
of Problem 46-1 will factor by drawing a square with rows and columns 
labeled by the wave functions which enter the secular equation, and indi- 
cating by zeros in the proper places in the square the vanishing matrix 
elements. 

46c. Reduction of Integrals. — Before discussing the conclusions 
which can be drawn from these equations, let us reduce somewhat 
further the integrals H uulf etc. The wave function II can be 
written in the form (Sec. 30a) 



fct = ^2(-"l) p Nl)a(lW2)a(2) Mc (3)^(3), 

p 



(46-12) 



Xm-46c] SLATER'S TREATMENT OF COMPLEX MOLECULES 371 

in which P represents a permutation of the functions u a a t etc., 
among the electrons. A typical integral can thus be expressed 
In the form 

Hum - ||2 2 ( ~ !)^^ f p ^ * (1)a(1)u * (2)a(2)w *< 3 >^( 3 > H 

P , P 

Pu a (l)a(l)u b (2)P(2)u e (3)a(3)dr. (46-13) 

Following exactly the argument of Section 30d for the atomic 
case, we can reduce this to the form 

Hum - 2(-l)W(l)"(l)<(2)a(2)^^ 
p 

ii»(2)0(2)u«(3)a(3)dr. (46-14) 

As in the atomic case, the integral vanishes unless the spins 
match, and there can be no permutation P which matches the 
spins unless 2m, is the same for II and III. In this case we 
see that the spins are matched for the permutations P which 
permute 123 into 132 or 231 so that only these terms contribute 
to the sum. When the spins match in an integral, the integration 
over the spin can be carried out at once, yielding the factor 
unity. We thus have the result 

K*(l) «(1) <(2) a(2) u c *(3) 0(3) J5M1) «(1) ti»(3) 0(3) u c {2) «(2) 
dr = /u*(l) w 6 *(2) u*(3) Hu a {\) u c (2) ti»(3) dr = (abc\H \acb), 

(46-15) 

in which we have introduced a convenient abbreviation, 
(abc\H\acb). 
In this way we obtain the following expressions: 

Hu - (abc\H\abc) - (abc\H\bac) - (abc\H\acb) 

- (abc\H\cba) + (abc\H\bca) + (dbc\H\cab), ] 
ffim - (abc\H\abc) - (abc\H\bac), 
Hmm - (a6c|H|o6c) - (ak|ff|<*a), 

Hiviv - (abc|ff|a6c) - (abc|ff|ac&), / < 46 ~ 16 ' 

Hum - (abc\H\cab) - (a6c|i?|ac6), 
Hmiv - (abc\H\cab) - (a6c|tf|6ac), 
Hinv - (abc\H\bca) - (a6c|ff|c&a). 

The expressions for the A's are the same with H replaced by 
unity. The integral (dbc\H\abc) is frequently called the Coulomb 
integral, because it involve? the Coulomb interaction of two 



372 THE STRUCTURE OF COMPLEX MOLECULES [XIII-46d 

distributions of electricity determined by u a , Ub t and u c . The 
other integrals such as (abc\H\bac) are called exchange integrals. 
If only one pair of orbitals has been permuted, the integral is 
called a single exchange integral; if more than one, a multiple 
exchange integral. If the orbital functions u a , u h} and u c were 
mutually orthogonal, many of these integrals would vanish, but 
it is seldom convenient to utilize orthogonal orbital functions in 
molecular calculations. Nevertheless, the deviation from orthog- 
onality may not be great, in which case many of the integrals 
can be neglected. 

46d. Limiting Cases for the System of Three Hydrogen Atoms. 
The values of the integrals #imi, etc., depend on the distances 
between the atoms a, 6, and c, and therefore the energy values 
and wave functions will also depend on these distances. It is 
interesting to consider the limiting case in which a is a large dis- 
tance from b and c, which are close together. It is clear that the 
wave function u a will not overlap appreciably with either u h or 
u ey so that the products u a Ub and u a u c will be essentially zero for 
all values of the coordinates. Such integrals as (abc\H\bac) 
will therefore be practically zero, and we can write 

#iui = #111111 = (abc\H\abc), 
Hui iv = #n iv = 0, 
#iviv = (dbc\H\abc) — (abc\H\acb), 
and 

#iim = ~ (abc\H\acb), 

thus obtaining the further relations 

H AA = (abc\H\abc) + (abc\H\acb), 
H BB = (abc\H\abc) - Y 2 {abc\H\acb) , 
and 

Has = -y 2 (abc\H\abc) - y 2 {abc\H\acb) . 

If we insert these values into the secular equation 46-10 we obtain 
as one of the roots the energy value 

W = ^> (46-17) 

and we find that the corresponding wave function is just the 
function A itself. 

It is found on calculation that exchange integrals involving 
orbitals on different atoms are usually negative in sign. In 



XIII-46d] SLATER'S TREATMENT OF COMPLEX MOLECULES 373 

case that such an integral occurs in the energy expression with a 
positive coefficient, it will contribute to stabilizing the molecule 
by attracting the atoms toward one another. Thus the expres- 
sion for H A a includes the Coulomb integral (abc\H\abc) and the 
exchange integral (abc\H\acb) with positive coefficient. Hence 
atoms b and c will attract one another, in the way corresponding 
to the formation of an electron-pair bond between them (exactly 
as in the hydrogen molecule alone). Similarly the function B 
represents the structure in which atoms a and c are bonded, and 
C that in which a and b are bonded. 

When we bring the three atoms closer together, so that all the 
interactions are important, none of these functions alone is the 
correct combination; they must be combined to give a wave 
function which represents the state of the system. Therefore 
when three hydrogen atoms are near together, it is not strictly 
correct to say that a certain two of them are bonded, while the 
third is not. 

We can, however, make some statements regarding the 
interaction of a hydrogen molecule and a hydrogen atom on the 
basis of the foregoing considerations. We have seen that when 
atom a is far removed from atoms b and c (which form a normal 
hydrogen molecule), the wave function for the system is function 
A. As a approaches b and c the wave function will not differ 
much from A, so long as the ab and ac distances are considerably 
larger than the be distance. An approximate value for the 
interaction energy will thus be H A a/&aa, with 

Haa — liiHnu + Hniui — 2Huiu) 
= (abc\H\abc) + (abc\H\acb) 

-y 2 (abc\H\bac) - V 2 {abc\H\cba) - (abc\H\cab), 

and a similar expression for Aaa- It is found by calculation that 
in general the single exchange integrals become important at 
distances at which the Coulomb integral and the orthogonality 
integral have not begun to change appreciably, and at which 
the multiple exchange integrals [(abc\H\cab) in this case] are still 
negligible. Thus we see that the interaction energy of a hydro- 
gen atom and a hydrogen molecule at large distances is 

-y 2 (abc\H\bac) - y 2 (abc\H\cba) . 

Each of these terms corresponds to repulsion, showing that the 
molecule will repel the atom. 



374 THE STRUCTURE OF COMPLEX MOLECULES [XIII-46e 

Approximate discussions of the interaction of a hydrogen atom 
and hydrogen molecule have been given by Eyring and Polanyi, 1 
and a more accurate treatment for some configurations has been 
carried out by Coolidge and James. 2 

46e. Generalization of the Method of Valence-bond Wave 
Functions. — The procedure which we have described above for 
discussing the interaction of three hydrogen atoms can be 
generalized to provide an analogous treatment of a system con- 
sisting of many atoms. Many investigators have contributed to 
the attack on the problem of the electronic structure of complex 
molecules, and several methods of approximate treatment have 
been devised. In this section we shall outline a method of treat- 
ment (due in large part to Slater) which may be called the 
method of valence-bond wave functions, without giving proofs of 
the pertinent theorems. The method is essentially the same as 
that used above for the three-hydrogen-atom problem. 

Let us now restrict our discussion to the singlet states of 
molecules with spin degeneracy only. For a system involving 
2n electrons and 2n stable orbitals (such as the Is orbitals in 2n 
hydrogen atoms), there are (2n)!/2 n n! different ways in which 
valence bonds can be drawn between the orbitals in pairs. Thus 
for the case of four orbitals a, 6, c, and d the bonds can be drawn 
in three ways, namely, 

a b a b a b 



d c d c d c 

A B C 

There are, however, only t , /'-m independent singlet wave 

functions which can be constructed from the 2n orbitals with 
one electron assigned to each Orbital (that is, with neglect of 
ionic structures). It was shown by Slater that wave functions 
can be set up representing structures A, B, and C, and that only 
two of them are independent. The situation is very closely 
analogous to that described in Section 466. 

1 H. Eyring and M. Polanyi, Naturwissenschaften, 18, 914 (1930); 
Z.f. phys. Chem. B12, 279 (1931). 
2 A. S. Coolidge and H. M. Jambs, /. Chem. Phys., 2, 811 (1934). 



Xin-46e] SLATER'S TREATMENT OF COMPLEX MOLECULES 375 

The very important observation was made by Rumer 1 that 
if the orbitals a, b } etc. are arranged in a ring or other closed 
concave curve (which need have no relation to the nuclear con- 
figuration of the molecule), and lines are drawn between orbitals 
bonded together (the lines remaining within the closed curve), 
the structures represented by diagrams in which no lines intersect 
are independent. These structures are said to form a canonical 
set. Thus in the above example the canonical set (correspond- 
ing to the order a, 6, c, d) comprises structures A and B. For 
six orbitals there are five independent structures, as shown 
in Figure 46-1. 



N* <S 



n 






m 



m 



%, 



<^ 






mis: 



Fig. 46-1. — The five canonical valence-bond structures for six orbitals, and 
some of their superposition patterns. 

The wave function corresponding to the structure in which 
orbitals a and 6, c and d, etc. are bonded is 



Po(l) 0(1) 6(2) «(2) c(3) 0(3) d(4) a(4) 



]• 



(46-18? 



in which P is the permutation operator described above (Sec. 
46c), and R is the operation of interchanging the spin functions 
a and p of bonded orbitals, such as a and b. The factor ( — 1) R 
equals +1 for an even number of interchanges and —1 for an 
odd number. The convention is adopted of initially assigning 
the spin function to orbital a, a to 6, etc. 

1 G. Rumer, Gottinger Nachr., p. 377, 1932. 



376 THE STRUCTURE OF COMPLEX MOLECULES [XIII-46e 

A simple method has been developed 1 of calculating the coeffi- 
cients of the Coulomb and exchange integrals in setting up the 
secular equation. To find the coefficient of the Coulomb 
integral for two structures, superimpose the two bond diagrams, 
as shown in Figure 46-1. The superposition pattern consists 
of closed polygons or islands, each formed by an even number of 
bonds. The coefficient of the Coulomb integral is 3^ n ~\ where 
i is the number of islands in the superposition diagram. Thus 
we obtain Hu = Q + • • • , Hm = }4Q + • • • , etc., in which 
Q represents the Coulomb integral (abed • • • \H\abcd • • • ). 

The coefficient of a single exchange integral such as 

(ab) = (abed • • • \H\bacd • • • ) 

is equal to //2 n ~*, in which /has the value — 3^ if the two orbitals 
involved (a and b) are in different islands of the superposition 
pattern; +1 if they are in the same island and separated by an 
odd number of bonds (along either direction around the polygon) ; 
and —2 if they are in the same island and separated by an even 
number of bonds. Thus we see that 

ffn = Q - }i(ac) + (ab) + • • • , H in = Q - 2(ac) + (ab) + 

• • • , etc. 

Let us now discuss the energy integral for a particular valence- 
bond wave function, in order to justify our correlation of valence- 
bond distribution and wave function as given in Equation 46-18. 
The superposition pattern for a structure with itself, as shown 
by I I in Figure 46-1, consists of n islands, each consisting of 
two bonded orbitals. We see that 

pn- _ ffii __ J_//i I "S^ /single exchange integrals for bonded\ 
An Au\ ^J\ pairs of orbitals / 

__ j/ %^ /single exchange integrals for non-bonded\ 
/2 -^— J \ pairs of orbitals / 

+ higher exchange integrals?. (46-19) 

It is found by calculation that the single exchange integrals 
are as a rule somewhat larger in magnitude than the other 
integrals. Moreover, the single exchange integral for two orbitals 

l L. Pauling, J. Chem. Phys. 1, 280 (1933). See also H. Eyring and 
G. E. Kimball, J. Chem. Phys. 1, 239 (1933), for another procedure. 



XIII-46f] SLATER'S TREATMENT OF COMPLEX MOLECULES 377 

on different atoms is usually negative in value for the interatomic 
distances occurring in molecules, changing with interatomic 
distance in the way given by a Morse curve (Sec. 35d). Those 
single exchange integrals which occur with the coefficient +1 in 
Equation 46-19 consequently lead to attraction of the atoms 
involved in the exchange, and the other single exchange integrals 
(with coefficient — }4) l ea d to repulsion; in other words, the 
wave function corresponds to attraction of bonded atoms and 
repulsion of non-bonded atoms and is hence a satisfactory wave 
function to represent the valence-bond structure under discussion. 

The valence-bond method has been applied to many problems, 
some of which are mentioned in the following section. It has 
been found possible to discuss many of the properties of the 
chemical bond by approximate wave-mechanical methods; an 
especially interesting application has been made in the treatment 
of the mutual orientation of directed valence bonds, 1 leading to 
the explanation of such properties as the tetrahedral orientation 
and the equivalence of the four carbon valences. 

46f. Resonance among Two or More Valence-bond Struc- 
tures. — It is found that for many molecules a single wave function 
of the type given in Equation 46-18 is a good approximation to 
the correct wave function for the normal state of the system j 
that is, it corresponds closely to the lowest root of the secular 
equation for the spin-degeneracy problem. To each of these 
molecules we attribute a single valence-bond structure, or 
electronic structure of the type introduced by G. N. Lewis, with 
two electrons shared between two bonded atoms, as representing 
satisfactorily the properties of the molecule. 

In certain cases, however, it is evident from symmetry or other 
considerations that more than one valence-bond wave function is 
important. For example, for six equivalent atoms arranged at 
the corners of a regular hexagon the two structures I and II of 
Figure 46-1 are equivalent and must contribute equally to the 
wave function representing the normal state of the system. 
It can be shown that, as an approximation, the benzene molecule 
can be treated as a six-electron system. Of the total of 30 valence 
electrons of the carbon and hydrogen atoms, 24 can be considered 

1 J. C. Slater, Phys. Rev. 37, 481 (1931); L. Pauling, J. Am. Chem. Soc. 63, 
1367 (1931); J. H. Van Vleck, /. Chem. Phys. 1, 177 (1933); R. Hultgrbn, 
Phys. Rev. 40, 891 (1932). 



378 



THE STRUCTURE OF COMPLEX MOLECULES [XIII -46f 



to be involved in the formation of single bonds between adjacent 
atoms, giving the structure 

H 



Hs 



C 

I 

c 



H 



/ 



V 



H 



^H 



H 

These single bonds use the Is orbital for each of the hydrogen 
atoms and three of the L orbitals for each carbon atom. There 
remain six L orbitals for the carbon atoms and six electrons, which 
can be represented by five independent wave functions corre- 
sponding to the five structures of Figure 46-1. We see that 
structures I and II are the Kekule* structures, with three double 
bonds between adjacent atoms, whereas the other structures 
involve only two double bonds between adjacent atoms. If, 
as an approximation, we consider only the Kekule* structures, 
we obtain as the secular equation 

Hi i - AnW #m - AiuW 
Hiu - &inW Hnu — kniiW 

in which also H u u = Hn and An n = Ai i. 
The solutions of this are 

Hi i + Hi n 



0, 



and 



W = 



W = 



An + Ai ii 
Hi i — Hi ii 



An 



Am 



the corresponding wave functions being ^i -f- ^ n and ^i — ^ n . 
Thus the normal state of the system is more stable than would 
correspond to either structure I or structure II. In agreement 
with the discussion of Section 41, this energy difference is called 
the energy of resonance between the structures I and II. 

As a simple example let us discuss the system of four equivalent 
univalent atoms arranged at the corners of a square. The two 
structures of a canonical set are 



XIII -46f J SLATER'S TREATMENT OF COMPLEX MOLECULES 379 



a b 


a b 




and 




d c 

I 


d c 
II 



If we neglect all exchange integrals of H except the single 
exchange integrals between adjacent atoms, which we call 
a [a = (ab) = (be) = (cd) = {da)}, and all exchange integrals 
occurring in A, the secular equation is found by the rules of 
Section 46e to be 



= 0. 



Q + a-W 1 AQ + 2a-y 2 W 

y 2 Q + 2a- y 2 w Q + a - w 

The solutions of this are W = Q + 2a and W = Q — 2a, of 
which the former represents the normal state, a being negative 
in sign. The energy for a single structure (I or II) is Wi = Q + a; 
hence the resonance between the two structures stabilizes the 
system by the amount a. 

Extensive approximate calculations of resonance energies foi 
molecules, especially the aromatic carbon compounds, have 
been made, and explanations of several previously puzzling 
phenomena have been developed. 1 Empirical evidence has 
also been advanced to show the existence of resonance among 
several valence-bond structures in many simple and complex 
molecules. 2 

It must be emphasized, as was done in Section 41, that the use 
of the term resonance implies that a certain type of approximate 
treatment is being used. In this case the treatment is based 
on the valence-bond wave functions described above, a procedure 
which is closely related to the systematization of molecule 
formation developed by chemists over a long period of years, 
and the introduction of the conception of resonance has per- 
mitted the valence-bond picture to be extended to include 

1 E. Huckel, Z. f. Phys. 70, 204 (1931), etc.; L. Pauling and G. W. 
Wheland, /. Chem. Phys. 1, 362 (1933); L. Pauling and J. Sherman, ibid. 
1, 679 (1933); J. Sherman, ibid. 2, 488 (1934); W. G. Penney, Proc. Roy. 
Soc. A146, 223 (1934); G. W. Wheland, J. Chem. Phys. 3, 230 (1935). 

2 L. Pauling, /. Am. Chem. Soc. 54, 3570 (1932); Proc. Nat. Acad. Sci. 
18, 293 (1932); L. Pauling and J. Sherman, /. Chem. Phys. 1, 606 (1933); 
G. W. Wheland, ibid, 1, 731 (1933); L. O. Brockway and L. Pauling, 
Proc. Nat. Acad. Sci. 19, 860 (1933). 



380 THE STRUCTURE OF COMPLEX MOLECULES [XIH-46g 

previously anomalous cases. A further discussion of this point 
is given in the following section. 

Problem 46-4. Set up the problem of resonance between three equivalent 
structures or functions fa, fai, ^m, assuming that Hn = Him = #111 in, 
etc. Solve for the energy levels and correct combinations, putting Ai i = 1 
and Ai n = 0. 

Problem 46-5. Evaluate the energy of a benzene molecule, considered 
as a six-electron problem: (a) considering only one Kekul6 structure; 
(b) considering both Kekule* structures; (c) considering all five structures. 
Neglect all exchange integrals of H except 

(ab) = (be) = (cd) = (de) = (ef) = (fa) = a, 

and all exchange integrals entering in A. 

46g. The Meaning of Chemical Valence Formulas. — The 

structural formulas of the organic chemist have been determined 
over a long period of years as a shorthand notation which 
describes the behavior of the compound in various reactions, 
indicates the number of isomers, etc. It is only recently that 
physical methods have shown directly that they are also fre- 
quently valid as rather accurate representations of the spatial 
arrangement of the atoms. The electronic theory of valence 
attempted to burden them with the additional significance of 
maps of the positions of the valence electrons. With the advent 
of quantum mechanics, we know that it is not possible to locate 
the electrons at definite points in the molecule or even to specify 
the paths on which they mov6. However, the positions of maxi- 
mum electron density can be calculated, and, as shown in Figure 
42-4, the formation of a bond does tend to increase the electron 
density in the region between the bonded atoms, which therefore 
provides a revised interpretation of the old concept that the 
valence electrons occupy positions between the atoms. 

The discussion of Section 466 shows that, at least in certain 
cases, the valence-bond picture can be correlated with an approxi- 
mate solution of the wave-mechanical problem. This correla- 
tion, however, is not exact in polyatomic molecules because 
functions corresponding to other ways of drawing the valence 
bonds also enter, although usually to a lesser extent. 

Thus the valence picture may be said to have a definite signifi- 
cance in terms of wave mechanics in those cases in which one 
valence-bond wave function is considerably more important than 
the others, but where this is not true the significance of the 



Xm-46h] SLATER'S TREATMENT OF COMPLEX MOLECULES 381 

structural formulas is less definite. Such less definite cases are 
those which can be described in terms of resonance. It is notable 
that the deficiency of the single structural formula in such cases 
has long been recognized by the organic chemist, who found that 
no single formula was capable of describing the reactions and 
isomers of such a substance as benzene. In a sense, the use of 
the term resonance is an effort to extend the usefulness of the 
valence picture, which otherwise is found to be an. imperfect 
way of describing the state of many molecules. 

46h. The Method of Molecular Orbitals. — Another method of 
approximate treatment of the electronic structure of molecules, 
called the method of molecular orbitals, has been developed and 
extensively applied, especially by Hund, Mulliken, andHiickel. 1 
This method, as usually carried out, consists in the approximate 
determination of the wave functions (molecular orbitals) and the 
associated energy values for one electron in a potential field corre- 
sponding to the molecule. The energy of the entire molecule is 
then considered to be the sum of the energies of all the electrons, 
distributed among the more stable molecular orbitals with no 
more than two electrons per orbital (Pauli's principle). A 
refinement of this method has been discussed in Section 436 in 
connection with the hydrogen molecule. 

As an example let us consider the system of four equivalent 
univalent atoms at the corners of a square, discussed in the 
previous section by the valence-bond method. The secular 
equation for a one-electron wave function (molecular orbital), 
expressed as a. linear combination of the four atomic orbitals 
u a > u b} u C) and u d , is 



= 0, 



- W P p 

P q - W P 

p q - W p 

P P q-W 

in which q is the Coulomb integral fu a (l) H'u a {l) dr and P is the 
exchange integral /u a (l) H'u b (l) dr for adjacent atoms, H' being 
the Hamiltonian operator corresponding to the molecular 

1 F. Hund, Z.f. Phys. 73, 1, 565 (1931-1932); R. S. Mulliken, /. Chem. 
Phys. 1, 492 (1933); etc.; J. E. Lennard-Jones. Trans. Faraday Soc. 25, 
668 (1929); E. Huckel, Z.f. Phys. 72, 310 (1931); 76, 628 (1932); 83, 632 
(1933); Trans. Faraday Soc. 30, 40 (1934). 



382 THE STRUCTURE OF COMPLEX MOLECULES [XIII-46h 

potential function assumed. We neglect all other integrals. 
The roots of this equation are 

Wi = q + 20, 

W 2 = q, 

Ws = q, 

W 4 = q - 20. 

Since is negative, the two lowest roots are Wi and W 2 (or TT 3 ), 
and the total energy for four electrons in the normal state is 

W = 2Wi + 2W 2 = 4? + 40. 

If there were no interaction between atoms a, 6 and c, d 
(corresponding to bond formation allowed only between a and b 
and between c and d), the energy for four electrons would still 
be 4g + 40. Accordingly in this example the method of molecu- 
lar orbitals leads to zero resonance energy. This is in poor 
agreement with the valence-bond method, which gave the 
resonance energy a. In most cases, however, it is found that 
the results of the two methods are in reasonably good agreement, 
provided that be given a value equal to about 0.6 a (for aromatic 
compounds). A comparison of the two methods of treatment 
has been made by Wheland. 1 It is found that the valence-bond 
method, when it can be applied, seems to be somewhat more 
reliable than the molecular-orbital method. On the other hand, 
the latter method is the more simple one, and can be applied to 
problems which are too difficult for treatment by the valence- 
bond method. 

Problem 4S-6. Treat the system of Problem 46-5 by the molecular- 
orbital method. Note that the resonance energy given by the two methods 
is the same if /3 = 0.553 a (using part c of Problem 46-5). 

1 G. W. Wheland, J. Chem. Phys. 2, 474 (1934). 



CHAPTER XIV 

MISCELLANEOUS APPLICATIONS OF QUANTUM 
MECHANICS 

In the following three sections we shall discuss four applications 
of quantum mechanics to miscellaneous problems, selected from 
the very large number of applications which have been made. 
These are: the van der Waals attraction between molecules 
(Sec. 47), the symmetry properties of molecular wave functions 
(Sec. 48), statistical quantum mechanics, including the theory 
of the dielectric constant of a diatomic dipole gas (Sec. 49), 
and the energy of activation of chemical reactions (Sec. 50). 
With reluctance we omit mention of many other important 
applications, such as to the theories of the radioactive decomposi- 
tion of nuclei, the structure of metals, the diffraction of electrons 
by gas molecules and crystals, electrode reactions in electrolysis, 
and heterogeneous catalysis. 

47. VAN DER WAALS FORCES 

The first detailed treatments of the weak forces between 
atoms and molecules known as van der Waals forces (which are 
responsible for the constant a of the van der Waals equation of 
state) were based upon the idea that these forces result from the 
polarization of one molecule in the field of a permanent dipole 
moment or quadrupole moment of another molecule, 1 or from 
the interaction of the permanent dipole or quadrupole moments 
themselves. 2 With the development of the quantum mechanics 
it has been recognized (especially by London 3 ) that for most 
molecules these interactions are small compared with another 
interaction, namely, that corresponding to the polarization of 
one molecule in the rapidly changing field due to the iiistan- 

1 P. Debye, Phys. Z. 21, 178 (1920); 22, 302 (1921). 

2 W. H. Keesom, Proc. Acad. Sci. Amsterdam 18, 636 (1915); Phys. Z. 22, 
129, 643 (1921). 

« F. London, Z. f. Phys. 63, 245 (1930). 

383 



384 MISCELLANEOUS APPLICATIONS [XIV-47a 

taneous configuration of electrons and nuclei of another mole- 
cule; that is, in the main the polarization of one molecule by the 
time-varying dipole moment of another. In the following sec 
tions we shall discuss the approximate evaluation of the energy 
of this interaction by variation and perturbation methods for 
hydrogen atoms (Sec. 47a) and helium atoms (Sec. 476), and then 
briefly mention the approximate semiempirical discussion for 
molecules in general (Sec. 47c). 

47a. Van der Waals Forces for Hydrogen Atoms. — For large 
values of the internuclear distance r A B — R the exchange phenom- 
enon is unimportant, and we can take as the unperturbed wave 
function for a system of two hydrogen atoms the simple product 
of two hydrogenlike Is wave functions, 

r = u U a(1) u U b(2). (47-1) 

The perturbation for this function consists of the potential 

energy terms 

p2, p" pi, p" 

H f = -— - — + — + — • (47-2) 

Tbi r A 2 r A B r l2 

Now this expression can be expanded in a Taylor's series in 
inverse powers of R = r A B, to give (with the two atoms located 
on the z axis) 

e 2 3 e 2 

H' = ^g(xix 2 + 2/12/2 - 2ziz 2 ) + 2R*l r * z * "~ r * Zl 

+ (2xiz 2 + 22/12/2 ~ 3siz 2 )(zi - z 2 )} 

+ l^M r l ~ 6rUl - $r\z\ - \bz\z\ 

+ 2(31*2 + 2/12/2 + 4z!Z 2 ) 2 J + • • • , (47-3) 

in which x if 1/1, «i are coordinates of the first electron relative to 
its nucleus, and x 2} 2/2, z 2 are coordinates of the second electron 
relative to its nucleus. The first term represents the interaction 
of the dipole moments of the two atoms, the second the dipole- 
quadrupole interaction, the third the quadrupole-quadrupole 
interaction, and so on. 

Let us first consider only the dipole-dipole interaction, using 
the approximate second-order perturbation treatment 1 of Section 

1 The first-order perturbation energy is zero, as can be seen from inspec- 
tion of the perturbation function. 



XIV-47a] VAN DER WAALS FORCES 385 

27e. It is necessary for us to evaluate the integral 

with H' given by 

H' = j^faxi + yiy2 - 2z l z i ). (47-4) 

It is seen that the cross-products in (i/') 2 vanish on integration, 
so that we obtain 



-£j>« 



or 

(ff'% = ~J ^*rJr,V«Wr = grfrj. (47-5) 

This expression, with r\ and r^ replaced by their value 3a§ (Sec. 
21c), gives, when introduced in Equation 27-47 together with 
Wq = —e 2 /a 0j the value for the interaction energy 

W' Q ' = -5?JJ?. (47-6) 

The fact that this value is also given by the variation method 
with the variation function ^°(1 + AH') shows that this is an 
upper limit for W'J (a lower limit for the coefficient of — e 2 al/R*). 
Moreover, by an argument similar to that of the next to the 
last paragraph of Section 27e it can be shown that the value 

— 8-p- is a lower limit to W' ' f so that we have thus determined 

the value of the dipole-dipole interaction to within about 15 
per cent. 

Variation treatments of this problem have been given by Slater 
and Kirkwood, 1 Hass6, 2 and Pauling and Beach. 3 It can be 
easily shown 4 that the second-order perturbation energy can 
be obtained by the use of a variation function of the form 

* = *°{l+ff , /(r 1 ,r t )}, 

with H' given by Equation 47-4. The results of the variation 

1 J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931). 

* H. R. Hasse, Proc. Cambridge Phil Soc. 27, 66 (1931). A rough treatment 
for various states has been given by J. Podolanski, Ann. d. Phys. 10, 695 
(1931). 

* L. Pauling and J. Y. Beach, Phys. Rev. 47, 686 (1935). 
4 This was first shown by Slater and Kirkwood. 



386 



MISCELLANEOUS APPLICATIONS 



[XIV-47a 



treatment for different functions /(ri, r 2 ) are given in Table 47-1. 
It is seen that the coefficient of —e 2 a\/R* approaches a value 1 
only slightly larger than 6.499; this can be accepted as very close 
to the correct value. 

So far we have considered only dipole-dipole interactions. 
Margenau 2 has applied the approximate second-order perturba- 
tion method of Section 27c to the three terms of Equation 47-3, 
obtaining the expression 



W" 



R* 



R* 



1416e 2 a 9 n 
R l 



+ 



• (47-7) 



It is seen that the higher-order terms become important at small 
distances. 

Table 47-1.— Variation Treatment of van der Waals Interaction 
of Two Hydrooen Atoms 

Variation function u lsA (l) tti„»(2)<l + --(xiz 2 + f/iJ/2 - 2z l z t )f(r i , r 2 )f 



/(ri, r 2 ) 


E -W° 


Reference* 


1. A 


--6.00e 2 a V^ 6 
-6 14 
-6 462 
-6 469 
-6 482 
-6 49 
-6 490 
-6 498 
-6 4984 
-6.49899 
-6.49903 


H 


2. - l Ar l r 2 /{r l -\- r 2 ) 


SK 


3. A +B(n + r 2 ) 

4. A + Brir 2 

5. A + B(n + r 2 ) + C ri r 2 . . 

6. Ar\r v 2 (v = 0.325) 

7. A + Br x r 2 +Ci\r\ 

8. A + Br x r 2 + C/JrJ + 1) r\r\ 

9. Polynomial f to r\r\ 

10. Polynomial to rjr^ 

11. Polynomial to r\r\ 


PB 

II 

PB 

SK 

II 

II 

PB 

PB 

PB 



* H = Hass6, SK = Slater and Kirkwood, PB - Pauling and Beach. 

f The polynomial contains all terms of degree 2 or less in n and 2 or less in ri. 



1 A straightforward but approximate application of second-order perturba- 
tion theory by R. Eisenschitz and F. London gave the value 6.47 for this 
coefficient [Z. /. Phys. 60, 491 (1930)]. The first attack on this problem 
was made by S. C. Wang, Phys. Z. 28, 663 (1927). The value found by 
him for the coefficient, 24 % 8 = 8.68, must be in error (as first pointed oul, 
by Eisenschitz and London), being larger than the upper limit 8 given above. 
The source of the error has been pointed out by Pauling and Beach, loc. cit. 

2 H. Margenau, Phys. Rev. 38, 747 (1931). More accurate values of the 
coefficients have been calculated by Pauling and Beach, loc. cit. 



XIV-47c] 



VAN DER WAALS FORCES 



387 



47b. Van der Waals Forces for Helium. — In treating the 
dipole-dipole interaction of two helium atoms, the expression for 
H ' consists of four terms like that of Equation 47-4, correspond- 
ing to taking the electrons in pairs (each pair consisting of an 
electron on one atom and one on the other atom). The variation 
function has the form 



* = rh + Xhwu, r,)l. 



Hass^ 1 has considered five variation functions of this form, 
shown with their results in Table 47-2. The success of his similar 
treatment of the polarizability of helium (function 6 of Table 
29-3) makes it probable that the value -lA13e 2 al/R« for W" 
is not in error by more than a few per cent. Slater and Kirk- 
wood 1 obtained values 1.13, 1.78, and 1.59 for the coefficient 
of — e 2 a\/R* by the use of variation functions based on their 
helium atom functions mentioned in Section 29e. An approxi- 
mate discussion of dipole-quadrupole and quadrupole-quadrupole 
interactions has been given by Margenau. 1 

Table 47-2. — Variation Treatment of van der Waals Interaction of 
Two Helium Atoms 



r 


f(nr 2 ) 


E - W° 


i. 

2. 
3. 
4. 
5. 


e -Z't 

e -z>» 

e- z ''(\ + 
e-*'«(l + 


CiU) 
CiU) 


A 

A + B ri r 2 
A + Br ir2 + Cr 
A 
A + Bt\ri 


V 2 


-1.079e 2 a 5 /^ 

-1.225 

-1.226 

-1.280 

-1.413 



47c. The Estimation of van der Waals Forces from Molecular 
Polarizabilities. — London 2 has suggested a rough method of 
estimating the van der Waals forces between two atoms or mole- 
cules, based on the approximate second-order perturbation 
treatment of Section 27e. We obtain by this treatment (see 
Sees. 27e and 29e) the expression 



2ne 2 z 2 



1 hoc. cit. 

* F. London, Z. f. Phys. 63, 245 (1930). 



(47-8) 



388 MISCELLANEOUS APPLICATIONS [XIV-48 

for the polarizability of an atom or molecule, in which n is the 
number of effective electrons, z* the average value of z 2 for these 
electrons (z being the coordinate of the electron relative to 
the nucleus in the field direction), and 7 the energy difference of 
the normal state and the effective zero point for energy, about 
equal in value to the first ionization energy. The van der Waals 
interaction energy may be similarly written as 

RVa + i B y ^ yj 

which becomes on introduction of a A and a B 

W = "2-gr 77+T.' ( 47 " 10 > 

or, in case the molecules are identical, 

With a in units 10~ 24 cm 3 and I in volt electrons, this is 

Dp 2 n*> 
W" — — ° 

R« ' 
in which 

D = 1.27a 2 /. 

It must be realized that this is only a very rough approximation. 
For hydrogen atoms it yields D = 7.65 (correct value 6.50) and 
for helium 1.31 (correct value about 1.4). 

For the further discussion of the validity of London's relation 
between van der Waals forces and polarizabilities, and of other 
applications of the relation, such as to the heats of sublimation 
of molecular crystals and the unactivated adsorption of gases by 
solids, the reader is referred to the original papers. 1 

48. THE SYMMETRY PROPERTIES OF MOLECULAR WAVE 

FUNCTIONS 

In this section we shall discuss the symmetry properties of 
molecular wave functions to the extent necessary for an under- 

1 F. London, loc. cit.; F. London and M. Polanyi, Z. f. phys. Chem, 11B, 
222 (1930); M. Polanyi, Trans. Faraday Soc. 28, 316 (1932); J. E. Lbnnabd- 
Jones. ibid, 28, 333 (1932). 



XIV-481 PROPERTIES OF MOLECULAR WAVE FUNCTIONS 389 

standing of the meaning and significance of the term symbols 
used for diatomic molecules by the spectroscopist. 

In Section 34 it was mentioned that the nuclear and electronic 
parts of an approximate wave function for a molecule can be 
separated by referring the electronic coordinates to axes deter- 
mined by the nuclear configuration. Let us now discuss this 
choice of coordinates for a diatomic molecule in greater detail. 
We first introduce the Cartesian coordinates X, Y, Z of the center 
of mass of the two nuclei relative to axes fixed in space, and the 
polar coordinates r, #, <p of nucleus A relative to a point midway 
between nucleus A and nucleus B as origin, 1 also referred to axes 




Fig. 48-1. — The relation between axes £, rj, f and X, Y, Z. 

fixed in space, as indicated in Figure 48-1. We next introduce 
the Cartesian coordinates £*, r) it £*» or the polar coordinates 
r if #i, <pi of each of the electrons, measured with reference, not 
to axes fixed in space, but instead to axes dependent on the 
angular coordinates # and <p determining the orientation of the 
nuclear axis. These axes, £, rj, f , are chosen in the following way. 
f is taken along the nuclear axis OA (Fig. 48-1), and £ lies in the 
XY plane, its sense being such that the Z axis lies between the 
rj and f axes (£, r), f forming a left-handed system, say). It is, 
moreover, often convenient to refer the azimuthal angles of all 
electrons but one to the azimuthal angle of this electron, using 
the coordinates <p h (p2 — <pi, <Pz — <pi, • • • in place of <pi, <p 2 , 
<Pz, • • • . 

1 It is convenient in this section to use these coordinates, which differ 
slightly from those adopted in Chapter X. 



390 MISCELLANEOUS APPLICATIONS [XIV-48a 

It has been shown 1 that these coordinates can be introduced 
in the wave equation, and that the wave functions then assume 
a simple form. We have discussed the wave function for the 
nuclear motions in detail in Chapter X. The only part of the 
electronic wave function which can be written down at once is 
that dependent on <pi. Inasmuch as the potential energy of the 
system is independent of <pi (as a result of our subterfuge of 
measuring the <p's of the other electrons relative to <pi), <pi is a 
cyclic coordinate, and occurs in the wave function only in the 
factor e ±iA *i, in which A can assume the values 0, 1, 2, • • • . 
The quantum number A thus determines the magnitude of the 
component of electronic orbital angular momentum along 
the line joining the nuclei. [A is somewhat analogous to the 
component Ml of the resultant orbital angular momentum 
(or azimuthal) quantum number L for atoms.] The value of A 
is expressed by the principal character of a molecular term 
symbol: 2 denoting A = 0; ft, A = ±1; A, A = ±2; etc. As 
in the case of atomic terms, the multiplicity due to electron spin 
is indicated by a superscript to the left, x 2 indicating a singlet, 
2 2 a doublet, etc. 

It may be mentioned that if we ignore the interactions of the 
electronic and nuclear motions the wave functions corresponding 
to A and —A correspond to identical energy values. This 
degeneracy is removed by these interactions, however, which 
lead to a small splitting of energy levels for A > 0, called A-type 
doubling. 2 The correct wave functions are then the sum and 
difference of those corresponding to A and —A. 

In the following sections we shall discuss the characteristic 
properties of diatomic molecules containing two identical nuclei 
(symmetrical diatomic molecules). 

48a. Even and Odd Electronic Wave Functions. Selection 
Rules. — By the argument of Section 40e we have shown that the 
transition probabilities for a diatomic molecule are determined 
in the main by the electric-moment integrals over the electronic 
parts of the wave functions, taken relative to the axes {, 17, f 
determined by the positions of the nuclei. Let us now classify 
the electronic wave functions of symmetrical diatomic molecules 

1 F. Hund, Z. f. Phys. 42, 93 (1927); R. deL. Kronig, ibid. 46, 814; 50, 
347 (1928); E. Wigner and E. E. Witmer, ibid. 51, 869 (1928). 
s See, for example, J. H. Van Vleck, Phys. Rev. 33, 467 (1929). 



XIV-48b] PROPERTIES OF MOLECULAR WAVE FUNCTIONS 391 

as even or odd, introducing the subscripts g (German gerade) 
for even terms and u (ungerade) for odd terms in the term symbols 
for identification. This classification depends on the behavior 
of the electronic wave function with respect to the transformation 
£1, Vi, f» — » —&, —Vi, —ft, that is, on inversion through the 
origin, even functions remaining unchanged by this operation, 
and odd functions changing sign. The argument of Section AQg 
leads to the following selection rule: Transitions are allowed 
only between even and odd levels (g —>u y u—tg). 

(Although electronic wave functions for diatomic molecules 
containing unlike nuclei cannot be rigorously classified as even 
or odd, they often approach members of these classes rather 
closely, and obey an approximate selection rule of the above 
type.) 

48b. The Nuclear Symmetry Character of the Electronic Wave 
Function. — We are now in a position to discuss the nuclear 
symmetry character of the electronic wave function for a diatomic 
molecule in which the nuclei are identical. Interchanging the 
two nuclei A and B converts # into t — & and <p into ir + <p; 
these coordinates, however, do not occur in the electronic wave 
function. The interchange of the nuclei also converts the 
coordinates (■»-, v%j f* of each electron into — &, rn 9 — f», and hence 
r if &i f <pi into r i} ir — & if tt — <pi [or <pi — <p x into — (^ — ^)]. 
In case that the electronic wave function is left unchanged by 
this transformation, the electronic wave function is symmetric 
in the nuclei; if the factor — 1 is introduced by the transformation, 
the electronic wave function is antisymmetric in the nuclei. 

The nuclear symmetry character of the electronic wave func- 
tion is represented in the term symbol by introducing the super- 
script + or — after taking cognizance of the presence of the 
subscript g or u discussed in Section 48a, the combinations 

g and u representing electronic wave functions symmetric in 

the nuclei, and g and u those antisymmetric in the nuclei. Thus 
we see that 

2+ and 2" are S N 
and 

27 and 2+ are A*. 

For A 9± there is little need to represent the symmetry char- 
acter in the term symbol, inasmuch as the S N and A N states 



392 



MISCELLANEOUS APPLICATIONS 



[XIV-48b 



occur in pairs corresponding to nearly the same energy value 
(A-type doubling), and in consequence the + and — superscripts 
are usually omitted. 

The states with superscript + are called positive states, and 
those with superscript — negative states. 

The principal use of the nuclear symmetry character is in 
determining the allowed values of the rotational quantum 
number X of the molecule. The complete wave functions for a 
molecule (including the nuclear-spin function) must be either 
symmetric or antisymmetric in the nuclei, depending on the 
nature of the nuclei involved. If the nuclei have no spins, 
then the existent functions are of one or the other of the types 
listed below. 

I. Complete wave function aS^ : 



+ 
9, 



X even < 



AX even - r , 
> u, K even 



AX odd 



AX odd 



+ ^ , , AX even - ~ , , 
u, X odd <- — > g, X odd 



II. Complete wave function A N : 



+ „ ,, AK even - r ,, 
g } X odd < > u, K odd 



AX odd 



K even < 



AX even 



AX odd 



+ g, K even 



It is seen that in either case the transitions allowed by the selec- 
tion rule g ** u are such that AX is even for H > — or > + 

transitions, and odd for -| > + or > — transitions. 

The selection rule AX = 0, ± 1 can be derived by the methods of 
Chapter XI ; this becomeb AX = for positive «*> negative transi- 
tions, and AX = ± 1 for positive — * positive or negative — > nega- 
tive transitions. 

In case that the nuclei possess spins, with spin quantum 
number /, both types of functions and transitions occur (the two 
not forming combinations), with the relative weights (/ + 1)// or 
//(/ + 1), as discussed in Section 43f. 



XIV-48b] PROPERTIES OF MOLECULAR WAVE FUNCTIONS 393 



Let us now consider a very simple example, in order to clarify 
the question; namely, the case of a molecule possessing only one 
electron, in the states represented by approximate wave functions 
which can be built from the four orbitals u A = s, p s , p x , p v about 
nucleus A, and four similar ones u B about nucleus B; s, p t , p x , p y 
being real one-electron wave functions such as given in Table 
21-4 for the L shell. We can combine these into eight functions 
of the form Sa + s B , s A — s B , etc. If the functions are referred 
to parallel axes for the two atoms and taken as in Table 21-4 



t 









Fig. 48-2. — Positive and negative regions of wave functions b, p M , p x , and p v 
for atoms A and B. 



except for a factor —1 for p tB (introduced for convenience), 
then they have the general nature shown in Figure 48-2, in which 
the functions u A + u B are designated, the plus and minus signs 
representing regions equivalent except for sign. From the 
inspection of this figure and a similar one for u A — u B (in which 
the signs are changed for u B ), it is seen that the eight functions 
have the following symmetry character in the nuclei: 



Function 


8 


V» 


V* 


Pv 


UA + u B 
ua — ub 


S» 
A» 


S N 
A N 


A* 

S" 


s« 

A* 



394 MISCELLANEOUS APPLICATIONS [XIV-48c 

By the argument given above we know that four of these are 
2 states, with A = 0, and four are II states. The II states are 
those formed from p z and p v (which are the linear combinations 
of the complex exponential functions p+ x and p-i). The two 
II states u A + u B are separated widely by the exchange integrals 
from the two u A — u B , and the A-type doubling will cause a 
further small separation of the nuclear-symmetric and nuclear- 
antisymmetric levels. The exchange terms similarly separate 
the u A + u B s and p z functions from the u A — u B functions. The 
best approximate wave functions would then be certain linear 
combinations of the two nuclear-symmetric functions and also 
of the two nuclear-antisymmetric functions. 

We can now write complete term symbols for the eight elec- 
tronic wave functions of our simple example, as follows: 

« Pz Vx Vv 

u A +u B *s+ 2 s; { 2 n+ 2 n-} 
u A -u B '2+ *x: pn+ m-} 

The identification as even or odd is easily made by inspection of 
Figure 48-2. The two 2 II W terms (one S N and one A N ) are placed 
in brackets to show that they form a A-type doublet, as are the 
two 2 U terms. 

48c. Summary of Results Regarding Symmetrical Diatomic 
Molecules. — The various symmetry properties which we have 
considered are the following: 

1. Even and odd electronic functions, indicated by subscripts 
g and u (Sec. 48a). Selection rule: Transitions allowed only 
between g and u. 

2. The nuclear symmetry of the complete wave function 
(including rotation of the molecule but not nuclear spin). Selec- 
tion rule: Symmetric-antisymmetric transitions not allowed. 

3. The nuclear symmetry of the electronic wave function, 

represented by the superscripts + and — , g and u being S N ; g 

and u, A N . Selection rule: AK = for positive-negative transi- 
tions, and AK = ± 1 f or positive-positive and negative-negative 
transitions. (This is not independent of 1 and 2. In practice 
1 and 3 are usually applied.) 

We are now in a position to discuss the nature of the spectral 
lines to be expected for a symmetrical diatomic molecule. We 



XIV-49] STATISTICAL QUANTUM MECHANICS 395 

have not treated the spin moment vector of the electrons, which 
combines with the angular momentum vectors A and K in various 
ways to form resultants; the details of this can be found in the 
treatises on molecular spectroscopy listed at the end of Chapter 
X. Let us now for simplicity consider transitions among x 2 
states, assuming that the nuclei have no spins, and that the 
existent complete wave functions are symmetric in the nuclei 
(as for helium). The allowed rotational states are then those 
with K even for *2+ and ! 2~, and those with K odd for *2~ 
and l 2+ 9 and the transitions allowed by 1 and 3 are the following: 

etc. 



etc. 



lv+ 


K = 




2 


4 




\/ N - 




i2+ 


K = 


1 




3 


1 2 + 

a 


K = 




2 


4 




1 




1 


i 


12- 


K = 




2 


4 


12" 


K = 


1 

! 




3 

I 


*2+ 


K = 


l 




3 


IV- 


K =■ 


l 




3 



etc. 



!2- K = 2 4 

49. STATISTICAL QUANTUM MECHANICS. SYSTEMS IN 
THERMODYNAMIC EQUILIBRIUM 

The subject of statistical mechanics is a branch of mechanics 
which has been found very useful in the discussion of the proper- 
ties of complicated systems, such as a gas. In the following 
sections we shall give a brief discussion of the fundamental 
theorem of statistical quantum mechanics (Sec. 49a), its applica- 
tion to a simple system (Sec. 496), the Boltzmann distribution 
law (Sec. 49c), Fermi-Dirac and Bose-Einstein statistics (Sec. 
49d), the rotational and vibrational energy of molecules (Sec. 49e), 
and the dielectric constant of a diatomic dipole gas (Sec. 49/). 
The discussion in these sections is mainly descriptive and 
elementary ; we have made no effort to carry through the difficult 
derivations or to enter into the refined arguments needed in a 



396 MISCELLANEOUS APPLICATIONS [XIV-49a 

thorough and detailed treatment of the subject, but have 
endeavored to present an understandable general survey. 

49a. The Fundamental Theorem of Statistical Quantum 
Mechanics. — Let us consider a large system with total energy 
known to lie in the range W to W + AW. We inquire as to the 
properties of this system. If we knew the wave function 
representing the system, values of the dynamical quantities 
corresponding to the properties of the system could be calculated 
by the methods of Section 12d. In general, however, there will 
be many stationary states of the system (especially if it be a very 
complicated system, such as a sample of gas of measurable 
volume) with energy values lying in the range W to W + AW y 
and our knowledge of the state of the system may not allow us to 
select one wave function alone as representing the system. 
Moreover, it might be possible for us to find a set of approximate 
wave functions for the system by ignoring weak interactions of 
parts of the system with each other or of the system and its 
environment; no one of these approximate wave functions 
would represent the state of the system over any appreciable 
period of time, and so we would not be justified in selecting any 
one of them for use in calculating values of dynamical quantities. 

Under these circumstances we might make calculations regard- 
ing the properties of the system for each of the wave functions 
with energy between W and AW, and then average the various 
calculations to obtain predictions regarding the average expected 
behavior of the system. The important question immediately 
arises as to what weights are to be assigned the various wave 
functions in carrying out this averaging. The answer to this 
question is given by the fundamental theorem of statistical 
quantum mechanics, as follows: In calculating average values of 
properties of a system with energy between W and AW y the same 
weight is to be assigned to every accessible wave function with 
energy in this range, in default of other information. (The wave 
functions are of course to be normalized and mutually orthog- 
onal.) This theorem can be derived from the equations of 
quantum mechanics (by methods such as the variation of con- 
stants, discussed in Chapter XI), with the aid of an additional 
postulate, 1 which is the quantum-mechanical analogue of the 

1 The postulate of randomness of phases. See, for example, W. Pauli, 
"Probleme der modernen Physik," S. Hirzel, Leipzig, 1928. 



XIV-49b] STATISTICAL QUANTUM MECHANICS 397 

ergodic hypothesis of classical statistical mechanics. We shall 
not discuss this derivation. 

The word accessible appears in the theorem for the following 
reason. If a system is known to be in one state at a given 
instant, and if it is known that it is impossible for any operative 
perturbation to cause a transition to a certain other state, then 
it is obviously wrong to include this latter state in the expression 
for the average. We have already met such non-combining 
states in our discussion of the symmetry of wave functions for 
collections of identical particles (Sees. 296, 30a). It was shown 
that if the system is known to be represented by a wave function 
symmetrical in all the identical particles composing it, no 
perturbation can cause it to change over to a state with an 
antisymmetrical wave function. The nature of the wave 
functions which actually occur is dependent upon the nature of 
the system. If it is composed of electrons or protons, the wave 
functions must be antisymmetric; if it is composed of hydrogen 
atoms, thought of as entities, the wave functions must be sym- 
metric in these atoms; etc. Moreover, we may sometimes have 
to take the passage of time into consideration in interpreting 
the word accessible. Let us consider as our system a helium 
atom, for example, which is known at the time t = to be in some 
excited singlet state, the wave function being symmetric in the 
positions of the electrons and antisymmetric in their spins. 
Transitions to triplet states can occur only as a result of perturba- 
tions affecting the electron spins; and, since these perturbations 
are very small, the probability of transition to all triplet states 
in a short time will be very small. In predicting properties for 
this system for a short period after the time t = 0, we would 
accordingly be justified in considering only the singlet states as 
accessible. 

49b. A Simple Application. — In order to illustrate the use of 
the fundamental theorem of statistical quantum mechanics, we 
shall discuss a very simple problem in detail. 

Let us consider a system composed of five harmonic oscillators, 
all with the same characteristic frequency v } which are coupled 
with one another by weak interactions. The set of product 
wave functions ^(0)^(6)^(0)^(^)^(6) can be used to construct 
approximate wave functions for the system by the use of the 
method of variation of constants (Chap. XI). Here ¥(a), • • • 



398 



MISCELLANEOUS APPLICATIONS 



[XIV-49b 



represent the harmonic oscillator wave functions (Sec. 11), 
the letters a, b, c, d f e representing the coordinates of the five 
oscillators. For each oscillator there is a set of functions ^n a (p) 
corresponding to the values 0, 1, 2, • • • for the quantum 
number n a . The total unperturbed energy of the system is 
Wl = (n« + V 2 )hv + . . • + (n, + V 2 )hv = (n + %)hr, in 
which n = n a + n b + n c + n d + n e . 

The application of the variation-of-constants treatment shows 
that if the system at one time is known to have a total energy 
value close to Wn>, where n f is a particular value of the quantum 
number n, then the wave function at later times can be expressed 
essentially as a combination of the product wave functions for 
n = n', the wave functions for n ^ n ! making a negligible 
contribution provided that the mutual interactions of the oscilla- 
tors are weak. Let us suppose that the system has an energy 
value close to \2}^hv 9 that is, that n' is equal to 10. The product 
wave functions corresponding to this value of n f are those 
represented by the 1001 sets of values of the quantum numbers 
tt 0> • • ' > n c given in Table 49-1. 

Table 49-1. — Sets of Quantum Numbers for Five Coupled Harmonic 
Oscillators with Total Quantum Number 10 



n a 


Tib 


n c 


rid 


n e 




10 


. 








Oetc. 


* (5) 


9 


. 1 


. 








(20) 


8 


. 2 











(20) 


7 


. 3 











(20) 


6 


. 4 











(20) 


5 


. 5 











(10) 


8 


. 1 


1 





-0 


(30) 


7 


. 2 


1 








(60) 


6 


. 3 


1 








(60) 


6 


. 2 


. 2 


. 





(30) 


5 


. 4 


. 1 


. 


. 


(60) 


5 


. 3 


. 2 


. 


. 


(60) 


4 


. 4 


2 


. 





(30) 


4 


. 3 


. 3 


. 





(30) 


7 


. 1 


. 1 


. 1 





(20) 



* The other sets indicated by " 
. . . 10 . 0, and 0. 
parentheses. 



n a rib n c rid 


n e 




6.2.1.1 


Oetc. 


(60) 


5.3.1.1 





(60) 


5.2.2.1 





(60) 


4 4.1.1 





(30) 


4.3.2.1 





(120) 


4.2.2.2 





(20) 


3.3.3.1 





(20) 


3.3.2.2 





(30) 


6.1.1.1 




(5) 


5.2.1.1 




(20) 


4.3.1.1 




(20) 


4.2 2.1 




(30) 


3.3 2.1 




(30) 


3.2.2.2 




(20) 


2 2.22 


2 


(1) 


e . 10 . . 


0, . 


. 10 . . 0, 


five, as shown 


by the 


number in 



In case that the interactions between the oscillators are of a 
general nature (the ab, ac f be, • • • interactions being different), 



XIV-49c] STATISTICAL QUANTUM MECHANICS 399 

all of the product functions will be accessible, and the funda- 
mental theorem then requires that over a long period of time the 
1001 product functions will contribute equally to the wave 
function of the system. In calculating the contribution of 
oscillator a, for example, to the properties of the system, we 
would calculate the properties of oscillator a in the states n a — 
[using the wave function ^o(a)L n « = 1, * * ' , n a = 10, and then 



0.4 



0.3 



Pn 02 



0.1 



0.0 



•23456789 10 II 

n a >- 

Fig. 49-1. — The probability values P n for system-part a in a system of five 
coupled harmonic oscillators with total quantum number n = 10 (closed circles), 
and values calculated by the Boltzmann distribution law (open circles). 

average them, using as weights the numbers of times that 
n a = 0, 1, 2, • • • , 10 occur in Table 49-1. These weights are 
given in Table 49-2. The numbers obtained by dividing by 
the total (1001) can be described as the probabilities that oscilla- 
tor a (or b, Cj • • • ) be in the states n a = 0, 1, 2, • • • , 10. 
These probability values are represented graphically in Figure 
49-1. 

49c. The Boltzmann Distribution Law. — We have been dis- 
cussing a system composed of a small number (five) of weakly 
interacting parts. A similar discussion (which we shall not 
give because it is necessarily rather involved) of a system com- 
posed of an extremely large number of »veakly interacting parts 
can be carried through, leading to a general expression for the 
probability of distribution of any one of the parts among its 



400 



MISCELLANEOUS APPLICATIONS 



[XIV-49C 



Tablk 49-2. — Weights fob States of Individual Oscillators in 
Coupled System 



n , etc. 


Weight 


Probability P„ a 





286 


0.286 


1 


220 


.220 


2 


165 


.165 


3 


120 


.120 


4 


84 


.084 


5 


56 


.056 


6 


35 


.035 


7 


20 


.020 


8 


10 


.010 


9 


4 


.004 


10 


1 


.001 


Total 


1001 


1.001 



stationary states. 1 The result of the treatment is the Boltzmann 
distribution law in its quantum-mechanical form: 

// all the product wave functions ^ (a) V(b) • • • of a system 
composed of a very large number of weakly interacting parts a, 
by * • • are accessible, then the probability of distribution of one 
of the parts, say a, among its states, represented by the quantum 
number n a , is given by the equation 

P na - Ae""*?, (49-1) 

in which W n is the energy of the part a in its various states and the 
constant A has such a value as to make 



X*\ - i- 



(49-2) 



There is considered to be one state for every independent wave 
function ^(a). The exponential factor, called the Boltzmann 
exponential factor, is the same as in the classical Boltzmann 
distribution law, which differs from Equation 49-1 only in the 
way the state of the system part is described. The constant k 
is the Boltzmann constant, with the value 1.3709 X 10 -16 erg 
deg -1 . The absolute temperature T occurring in Equation 49-1 

1 That is, among the stationary states for this part of the system when 
isolated from the other parts. 



XIV-49C] STATISTICAL QUANTUM MECHANICS 401 

is introduced in the derivation of this equation by methods 
closely similar to those of classical statistical mechanics. 

Some indication of the reasonableness of this equation is given 
by comparing it with the results of our discussion of the system of 
five coupled harmonic oscillators. The open circles in Figure 
49-1 represent values of P n<j calculated by Equation 49-1, with 
kT placed equal to %hv (this leading approximately to the average 
value Yilftv for W n<x , as assumed in the earlier discussion). It is 
seen that there is general agreement, the discrepancies arising 
from the fact that the number of parts of the system (five) is 
small (rather than very large, as required in order that the 
Boltzmann distribution law be applicable). 

In Equation 49-1 each wave function is represented separately. 
It is often convenient to group together all wave functions 
corresponding to the same energy, and to write 

Pi = Ap x e kT , (49-3) 

in which pi is the degree of degeneracy or a priori probability or 
quantum weight of the energy level Wi. 

In case that the wave functions for the part of the system 
under consideration are very numerous and correspond to energy 
values lying very close together, it is convenient to rewrite the 
distribution law in terms of P(W), such that P(W)dW is the 
probability that the energy of the system part lie between W 
and W + dW, in the form 

_w_ 
P{W) = Ap(W)e kT , (49-4) 

in which p(W)dW is the number of wave functions for the system 
part in the energy range W to W + dW. 

As an illustration of the use of Equation 49^i let us consider 
the distribution in translational energy of the molecules of a 
gas (the entire gas being the system and the molecules the system 
parts) such that all product wave functions are accessible. 1 
It is found (by the use of the results of Section 14, for example) 
that p(W) is given by the equation 

vW) = ^!!p, (49-5) 

1 We shall see in the next section that actual gases are not of this type. 



402 MISCELLANEOUS APPLICATIONS [XIV-49d 

in which V is the volume of the box containing the gas and m 
is the mass of a molecule. The Maxwell distribution law for 
velocities is obtained by substituting this in Equation 49-4 and 
replacing W by y^mv 2 , v being the velocity of the molecule. 

Problem 49-1. Derive Equation 49-5 with the use of the results of 
Section 14. By equating W to the kinetic energy ]^mv 2 (v being the 
velocity), derive the Maxwell distribution law for velocities, and from it 
calculate expressions for the mean velocity and root-mean-square velocity 
of gas molecules. 

It will be shown in the following section that the Boltzmann 
distribution law is usually not strictly applicable in discussing 
the translational motion of molecules. 

49d. Fermi-Dirac and Bose-Einstein Statistics. — As stated in 
the foregoing section, the Boltzmann distribution law is applicable 
to the parts of a system for which all product wave functions are 
accessible. The parts of such a system are said to conform to 
Boltzmann statistics. Very often, however, we encounter systems 
for which not all product wave functions are accessible. We 
have seen before (Sec. 29, etc.) that the wave functions for a 
system of identical particles can be grouped into non-combining 
sets of different symmetry character, one set being completely 
symmetric in the coordinates of the particles, one completely 
antisymmetric, and the others of intermediate symmetry char- 
acter. Only the wave functions of one symmetry character are 
accessible to a given system of identical particles. 

Thus our simple system of five harmonic oscillators would be 
restricted to wave functions of one symmetry character if the 
interactions ab y ac } 6c, • • • were equivalent, that is, if the 
oscillators were identical. 1 It was to avoid this that we made 
the explicit assumption of non-equivalence of the interactions in 
Section 4%. The accessible wave functions for five identical 
oscillators would be the completely symmetric ones, the com- 

1 In order for the oscillators to behave identically with respect to external 
perturbations as well as mutual interactions they would have to occupy the 
same position in space; that is, to oscillate about the same point. A 
system such as a crystal is often treated approximately as a set of coupled 
harmonic oscillators (the atoms oscillating about their equilibrium posi- 
tions). The Boltzmann statistics would be used for this set of oscillators, 
inasmuch as the interactions depend on the positions of the oscillators in 
space in such a way as to make them non-identical. 



XIV-49dJ STATISTICAL QUANTUM MECHANICS 403 

pletely antisymmetric ones, or those with the various inter- 
mediate symmetry characters. It is only the two extreme types 
which have been observed in nature. There are 30 completely 
symmetric wave functions for n = 10; they are formed from the 
successive sets in Table 49-1 by addition, the first being 

~{ (10.0.0.0.0) + (0.10.0.0.0) + (0.0.10.0.0) + (0.0.0.10.0) + 

(0.0.0.0.10)) 

and the last being (2.2.2.2.2). From these we can obtain 
weights for the successive values, similar to those given in 
Table 49-2; these weights will not be identical with those of the 
table, however, and so will correspond to a new statistics. This 
is very clearly seen for the case that only the completely anti- 
symmetric wave functions are accessible. The only wave 
function with n = 10 which is completely antisymmetric is that 
formed by suitable linear combination of the 120 product func- 
tions (4.3.2.1.0), etc., marked A in Table 49-1 (the other functions 
violate Pauli's principle, the quantum numbers not being all 
different). Hence even at tjie lowest temperatures only one of 
the five oscillators could occupy the lowest vibrational state, 
whereas the Boltzmann distribution law would in the limit 
T — » place all five in this state. 

If only the completely antisymmetric wave functions are accessible 
to a system composed of a large number of weakly interacting parts, 
the system parts conform to the Fermi-Dirac statistics; 1 if only the 
completely symmetric wave functions are accessible, they conform 
to the Bose-Einstein statistics. 2 

The Fermi-Dirac distribution law in the forms analogous to 
Equations 49-1, 49-3, and 49-4 is 

P n = —^ > (49-6) 

Ae kT +N 

1 E. Fermi, Z. f. Phys. 36, 902 (1926); P. A. M. Dirac, Proc. Roy. Soc. 
A112, 661 (1926). This statistics was first developed by Fermi, on the basis 
of the Pauli exclusion principle, and was discovered independently by Dirac, 
using antisymmetric wave functions. 

2 S. N. Bosb, Z.f. Phys. 26, 178 (1924); A. Einstein, Sitzber. Preuss. Akad. 
Wiss. p. 261, 1924; p. 3, 1925. Bose developed this statistics to obtain a 
formal treatment of a photon gas, and Einstein extended it to the case of 
material gases. 



404 MISCELLANEOUS APPLICATIONS [XIV-49d 

Pi = —^ > (49-7) 

Ae kT +N 
and 

P(W) = y°-j (49-8) 

in each of which the constant A has such a value as to make the 
sum or integral of P equal to unity. Here N is the number of 
identical system parts in which the accessible wave functions 
are antisymmetric. 

Problem 49-2. Show that at very low temperatures the Fermi-Dirac 
distribution law places one system part in each of the N lowest states. 

The Fermi-Dirac distribution law for the kinetic energy of the 
particles of a gas would be obtained by replacing p(W) by the 
expression of Equation 49-5 for point particles (without spin) 
or molecules all of which are in the same non-degenerate state 
(aside from translation), or by this expression multiplied by the 
appropriate degeneracy factor, which is 2 for electrons or protons 
(with spin quantum number Yi) ) or in general 27 + 1 for spin 
quantum number I. This law can be used, for example, in dis- 
cussing the behavior of a gas of electrons. The principal 
application which has been made of it is in the theory of metals, * 
a metal being considered as a first approximation as a gas of 
electrons in a volume equal to the volume of the metal. 

Problem 49-3. (a) Evaluate the average kinetic energy of the valence 
electrons (ignoring the K electrons and the nuclei) in a crystal of lithium 
metal at 0°A, and discuss the distribution of energy, (b) Calculate the 
number of electrons at 298 C A with kinetic energy 0.10 v.e. greater than the 
maximum for 0°A. The density of lithium is 0.53 g./cm 3 . 

The Bose-Einstein distribution law in the forms analogous to 
Equations 49-6, 49-7, and 49-8 is 

P« = — ^ 1 (49-9) 

Ae kT - N 

* W. Patjli, Z. /. Phys. 41, 81 (1927); A. Sommerfeld, Z. /. Phys. 47, 1, 
43 (1928); etc. Review articles have been published by K. K. Darrow, 
Rev. Mod. Phys. 1, 90 (1929); J. C. Slater, Rev. Mod. Phys. 6, 209 (1934); 
etc. 



XIV-49e] STATISTICAL QUANTUM MECHANICS 405 

Pi = — ^p > (4&-10) 

Ae** - N 
and 

P(W) = -?£P—> (49-11) 

Ae* r - N 

in which the symbols retain their former significance. The Bose- 
Einstein statistics is to be used for photons, 1 deuterons, helium 
atoms, hydrogen molecules, etc. 

For many systems to which Fermi-Dirac or Bose-Einstein 
statistics is to be applied the term ±N is negligible compared to 

J? 
Ae kT , and the appropriate equations are very closely approxi- 
mated by the corresponding Boltzmann equations. Thus helium 
gas under ordinary conditions shows no deviations from the 
perfect gas laws (Boltzmann statistics) which can be attributed 
to the operation of Bose-Einstein statistics. At very low 
temperatures and very high pressures, deviations due to this 
cause should occur, however; this degeneration 2 has not been 
definitely shown to occur for material gases by experiment, 3 
the principal difficulty being that real gases elude investigation 
under extreme conditions by condensing to a liquid or solid 
phase. 

49e. The Rotational and Vibrational Energy of Molecules. — 
In the statistical discussion of any gas containing identical 
molecules, cognizance must be taken of the type of statistics 
applicable. Often, however, we are not primarily interested in 
the translational motion of the molecules but only in their dis- 
tribution among various rotational, vibrational, and electronic 
states. This distribution can usually be calculated by the use 
of the Boltzmann distribution law, the effect of the symmetry 
character being ordinarily negligible (except in so far as the sym- 

1 With appropriate modifications to take account of the vanishing rest 
mass of photons. 

* The word degeneracy is used in this sense (distinct from that of Section 
14), the electrons in a metal being described as constituting a degenerate 
electron gas. 

* G. E. Uhlenbeck and L. Gboppeb, Phys. Rev. 41, 79 (1932), and refer- 
ences there quoted. 



406 MISCELLANEOUS APPLICATIONS [XIV-49e 

metry character relative to identical particles in the same 
molecule determines the allowed wave functions for the molecule). 

In case that the energy of a molecule can be represented as the 
sum of several terms (such as rotational, vibrational, electronic, 
and translational energy), the Boltzmann factor can be written 
as the product of individual Boltzmann factors, and the con- 
tributions of the various energy terms to the total energy of the 
system in thermodynamic equilibrium and to the heat capacity, 
entropy, and other properties can be calculated separately. To 
illustrate this we shall discuss the contributions of rotational and 
vibrational motion to the energy content, heat capacity, and 
entropy of hydrogen chloride gas. 

As shown in Chapter X, the energy of a hydrogen chloride 
molecule in its normal electronic state can be approximately 
represented as 

W VtK = (v + V 2 )hv + K(K + 1)J^, (49-12) 

in which v is the vibrational frequency, I the moment of inertia 
of the molecule, and v and K the vibrational and rotational 
quantum numbers, with allowed values v = 0, 1, 2, • • • and 
K = 0, 1, 2 • • • . At all but very high temperatures the 
Boltzmann factor for excited electronic states is very small, so 
that only the normal electronic state need be considered. Using 
Equation 49-3, we write for the probability that a molecule be 
in the state v,K the expression 







PvK = PvPk 


(49-13) 


in which 




(»+K)*» 








P, = Be kT , 


(49-14) 


and 




K(K+l)h* 






Pk 


= C(2K + l)e »"»* , 


(49-15) 



2K + 1 being the quantum weight of the Kth rotational state. 
B and C have values such that 

00 00 

%P V = 1 and 2 P * = L 

»-0 JK>0 

It is seen that the average rotational and vibrational energy 
per molecule can hence be written as 



XTV-49e] STATISTICAL QUANTUM MECHANICS 



407 



oo oe 



W = 22 P ' Pjt { (1 ' + l/i)hv + K{K + 1} 8^?}' 

or, since the summation over K can be at once carried out for the 
first term (to give the factor 1) and that over v for the second 
term, 

with 



w vlhT . = Jo* + y 2 )hvP, 



and 



that is, the average energy is separable into two parts in the 
same way as the energy W v ,k (Eq. 49-12). By introducing the 
variables 1 

hv 



kT 



h 2 



(49-16) 



8ir 2 IkT 



these parts can be written as 

00 

%(v + y 2 )xe~^W* 



JFvibr. = kT 



» = 



and 



J K(K + 1)(2K + l)<re-*<*+i>r 



(49-17) 



W K t, = kT 



K~0 



%(2K + 1)€ 



(49-18) 



-JT(JM-l)* 



25T = 



l The symbol <r is conventionally used in this way as well as for the 
quantity A 2 /8r 2 /, as in Section 35. 



408 MISCELLANEOUS APPLICATIONS [XIV-4W 

the sums in the denominators corresponding to the factors JB 
and C of Equations 49-14 and 49-15. Expressions for the 
vibrational and rotational heat capacity C V ibr. and C ro t. can be 
obtained by differentiating with respect to T y and the contribu- 
tions of vibration and rotation to the entropy can then be 

obtained as S vib , = f ^'dT and S rot . = f ^fdT. 

Problem 49-4. Considering only the first two or three excited states, 
calculate the molal vibrational energy, heat capacity, and entropy of 
hydrogen chloride at 25°C, using the vibrational wave number v — 
2990 cm" 1 . 

Problem 49-6. By replacing the sums by integrals, show that the 
expressions 49-17 and 49-18 approach the classical value kT for large T. 

Problem 49-6. Calculate the rotational energy curve (as a function of T) 
for hydrogen chloride at temperatures at which it begins to deviate from 
zero. The internuclear distance is 1.27 A. 

The treatment of ortho and para hydrogen, mentioned in 
Section 43/, differs from that of hydrogen chloride only in the 
choice of accessible rotational wave functions. For para hydro- 
gen K can assume only the values 0, 2, 4, • • • , the quantum 
weight being 2K + 1. For ortho hydrogen K can have the 
values 1, 3, 5, • • • , with quantum weight S(2K + 1), the 
factor 3 being due to the triplet nuclear-spin functions. Ordinary 
hydrogen is to be treated as a mixture of one-quarter para and 
three-quarters ortho hydrogen, inasmuch as only the states with 
K even are to be considered as accessible to the para molecules, 
and those with K odd to the ortho molecules. In the presence 
of a catalyst,- however, all states become accessible, and the gas 
is to be treated as consisting of molecules of a single species. 

Problem 49-7. Discuss the thermodynamic properties (in their depend- 
ence on rotation) of the types of hydrogen mentioned above. 

Problem 49-8. Similarly treat deuterium and protium-deuterium 
molecules (see footnote, Sec. 43/). 

49f. The Dielectric Constant of a Diatomic Dipole Gas. — 

Under the influence of an electric field, a gas whose molecules 
have a permanent electric moment and in addition can have a 
further moment induced in them by electronic polarization 
becomes polarized in the direction of the field, the polarization 
per unit volume being 



XIV-49f] STATISTICAL QUANTUM MECHANICS 409 

P = g jqrg*' = N* + NaF > (49-19) 

in which € is the dielectric constant of the gas, F the strength of 
the applied field (assumed to be parallel to the z axis), N the 
number of molecules in unit volume, and a the polarizability of 
the molecule. JT Z represents the average value of JT t for all 
molecules in the gas, JT Z being the average value of the z compo- 
nent of the permanent electric moment /x of a molecule in a given 
state of motion. It was shown by Debye 1 that according to 
classical theory ]T Z has the value 

JT. - ^ (49-20) 

We shall now show that for the special case of a diatomic dipole 
gas, such as hydrogen chloride, the same expression is given 
by quantum mechanics. 

Let us consider that the change of the permanent moment 
fjL with change in the vibrational quantum number v can be 
neglected. ]T Z is then given by the equation 

f z = *2fKuWu(KM), (49-21) 

K,M 

in which 2 

Pkm = Ae- KiK + l) *, (49-22) 

with cr = h 2 /8ir 2 IkT> as in Equation 49-16. Our first task is 
hence to evaluate 'JT Z {KM) 1 which is the average value of 

fX z = n cos # 

for a molecule in the rotational state described by the quantum 
numbers K and M, & being the angle between the moment /x of 
the molecule (that is, the nuclear axis) and the z axis. 
The value of JT Z (KM) is given by the integral 

Wz{KM) = S+* M p cos tty*A,dr, (49-23) 

in which ^ K m is the first-order perturbed wave function for the 
molecule in the electric field. It is found, on application of the 

1 P. Debye, Phys. Z. 13, 97 (1912). 

* It is assumed at this point that the energy of interaction of the molecule 
and the field can be neglected in the exponent of the Boltzmann factor. 
An investigation shows that this assumption is valid 



410 MISCELLANEOUS APPLICATIONS [XIV-49f 

usual methods of Chapters VI and VII, the perturbation function 
being 

H' = -fjiFcoad, (49-24) 

that jr x (KM) has the value 

-r KM \ - 8t2I » 2F & M2 - g(g + l l \ awn 

H^am) - h2 (2X^ 1)K(K + 1)(2K + S) W- M > 

(see Prob. 49-9). 

Inasmuch as P K m is independent of the quantum number M , 
to the degree of approximation of our treatment, we can at once 
calculate the average value of jT z (KM) for all states with the same 
value of K y by summing jr,(KM) for M = —K,K + 1, • • • , 
+K, and dividing by 2K + 1. The only part of 49-25 which 

+ K 

involves any difficulty is that in M 2 . The value of V M 2 

M--K 

is HK(K + 1)(2K + 1); using this, we see that 

+ K 

UK) = gA-j ^ ^ M ) = °> * > °- ( 49 ~ 26 ) 

M--K 

Thus we have obtained the interesting result that the only 
rotational state which contributes to the polarization is that with 
K = 0. The value of JT t for this state is seen from Equation 
49-25 to be 

WM = ^g?> (49-27) 

and fit hence is given by the equation 

= . gP __^__ ., (49 _ 28) 



c 2 (2K + 1)€ 



2T-0 



in which the sum in the denominator corresponds to the constant 
A of Equation 49-22. For small values of <r (such as occur in 
actual experiments) this reduces to 



= _ n 2 F 
M * " ZkT' 



(49-29) 



which is identical with the classical expression 49-20. On 
introduction in Equation 49-19, this gives the equation 



XIV-49fl STATISTICAL QUANTUM MECHANICS 411 

Problem 49-9. Using the surface-harmonic wave functions mentioned 
in the footnote at the end of Section 35c, derive Equation 49-25, applying 
either the ordinary second-order perturbation theory or the method of 
Section 27a. 

Problem 49-10. Discuss the approximation to Equation 49-28 provided 
by 49-29 for hydrogen" chloride molecules (m ~ 1.03 X 10"" 18 e.s.u.) in a 
field of 1000 volts per centimeter. 

It can be shown 1 that Equation 49-30 is not restricted to dia- 
tomic molecules in its application, but is valid in general, except 
for a few special cases (as, for example, for a molecule with 
electric moment largely dependent on the vibrational state, or 
on the state of rotation of one part of the molecule about a single 
bond, etc.). With the use of this equation the electric moments 
of molecules can be determined from measurements on the 
temperature coefficient of the dielectric constants of gases and 
dilute solutions and in other ways. This has been done for a very 
large number of substances, with many interesting structural 
conclusions. An illustration is the question of which of the two 
isomers of dichlorethylene is the cis and which the trans form, 
i.e., which compound is to be assigned to each of the formulas 
shown below: 

H H 

\ / 

C=C 

/ \ 

CI CI 

cis form 

The trans form is symmetrical and therefore is expected to have 
zero electric moment. It is found experimentally that the 
compound which the chemists had previously selected as the 
trans form does in fact have zero moment, whereas the cis form 
has a moment of about 1.74 X 10~ 18 e.s.u. (The unit 10 -18 e.s.u. 
is sometimes called a Debye unit.) Strong evidence for the 
plane structure of benzene is also provided by electric-moment 
data, and many other problems of interest to chemists have been 
attacked in this way. 

1 See the references at the end of the section, in particular Van Vleck. 



H 


CI 


\ 


/ 


C= 


=C 


/ 


\ 


CI 


H 


trans form 



412 MISCELLANEOUS APPLICATIONS [XIV-50 

An equation which is very closely related to Equation 49-30 is 
also applicable to the magnetic susceptibility of substances; 
indeed, this equation was first derived (by Langevin 1 in 1905) 
for the magnetic case. The temperature-dependent term in this 
case corresponds to paramagnetism, fx representing the magnetic 
moment of the molecule; and the other term, which in the mag- 
netic case is negative, corresponds to diamagnetism. For 
discussions of the origin of diamagnetism, the composition of the 
resultant magnetic moment /x from the spin and orbital moments 
of electrons, etc., the reader is referred to the references men- 
tioned below. 

References on Magnetic and Electric Moments 

J. H. Van Vleck: "The Theory of Electric and Magnetic Suscepti- 
bilities," Oxford University Press, 1932. 

C. P. Smyth: "Dielectric Constant and Molecular Structure, ,, Chemical 
Catalog Company, Inc., New York, 1931. 

P. Debye: "Polar Molecules," Chemical Catalog Company, Inc., New 
York, 1929. 

E. C. Stoner: "Magnetism and Atomic Structure," E. P. Dutton & 
Co., Inc., New York, 1926. 

The most extensive table of values of dipole moments available at present 
is that given in an Appendix of the Transactions of the Faraday Society, 1934. 

General References on Statistical Mechanics 

R. C. Tolman: "Statistical Mechanics with Applications to Physics and 
Chemistry," Chemical Catalog Company, Inc., New York, 1927. 

R. H. Fowler: "Statistical Mechanics," Cambridge University Press, 
1929. 

L. Brillotjin: "Les Statistiques Quantiques," Les presses universitaires de 
France, Paris, 1930. 

K. K. Darrow: Rev. Mod. Phys. 1, 90 (1929). 

R. H. Fowler and T. E. Sterne: Rev. Mod. Phys. 4, 635 (1932). 

50. THE ENERGY OF ACTIVATION OF CHEMICAL REACTIONS 
A simple interpretation of the activation energy E of a chemical 
reaction such as 

A + BC -> AB + C (50-1) 

is provided by the assumption that the molecule BC in its normal 
electronic state is not able to react with the atom A, and that 
reaction occurs only between A and an electronically excited 
molecule BC*, E being then the energy difference of the normal 

1 P. Langbvin, J. de phys. 4, 678 (1905). 



XIV-60J ENERGY OF ACTIVATION OF CHEMICAL REACTIONS 413 

and the excited molecule. A reasonable alternative to this 
was given in 1928 by London, 1 who suggested that such a reaction 
might take place without any change in the electronic state of 
the system (other than that accompanying the change in the 
internuclear distances corresponding to the reaction 50-1, as 
discussed in Section 34). The heat of activation would then be 
obtained in the following way. We consider the electronic 
energy TF (£) for the normal electronic state of the system as a 



'AB 




'BC 

Fig. 50-1. — The electronic energy surface (showing contour lines with 
increasing energy 1, 2, 3, etc.) for a system of three atoms arranged iinearly, as a 
function of the internuclear distances tab and tbc. 

function of the nuclear coordinates f . W (%) will have one value 
for the nuclear configuration in which nuclei B and C are close 
together, as in the normal molecule BC, and A is far removed, 
and another value for the AB + C nuclear configuration. (The 
difference of these, corrected for the energy of oscillation and 
rotation of the molecules, is the energy change during the 
reaction.) Now in order to change from one extreme configura- 
tion to the other, the nuclei must pass through intermediate 
configurations, as atom A approaches B and C recedes from it, 
and the electronic energy Wo(£) would change with change in 

1 F. London in the Sommerfeld Festschrift, "Probleme der modernen 
Phyaik/'p. 104, S. Hirzel, Leipzig, 1928. 



414 MISCELLANEOUS APPLICATIONS [XIV-50 

configuration, perhaps as shown in Figure 50-1. The change 
from A + BC, represented by the configuration point P, to 
AB + C, represented by the configuration point P", could take 
place most easily along the path shown by the dotted line. 
We have seen in Section 34 that the electronic energy can be 
treated as a potential function for the nuclei ; it is evident that in 
order for reaction to take place the nuclei must possess initially 
enough kinetic energy to carry them over the high point P' of 
the saddle of the potential function of Figure 50-1. The energy 
difference Wo(P') — Wo(P) t after correction for zero-point 
oscillational energy, etc., would be interpreted as the activation 
energy E. 

No thoroughly satisfactory calculation of activation energies in 
this way has yet been made. The methods of treatment dis- 
cussed for the hydrogen molecule in Section 43, in particular the 
method of James and Coolidge, could of course be extended to a 
system of three protons and three electrons to provide a satis- 
factory treatment of the reaction H + H 2 — > H2 + H. This 
calculation would be difficult and laborious, however, and has 
not been carried out. Several rough calculations, providing 
values of E for comparison with the experimental value 1 of 
about 6 kcal/mole (from the ortho-para hydrogen conversion), 
have been made. In Section 46c? we have seen that at large 
distances the interaction of a hydrogen atom A and a hydrogen 
molecule BC is given approximately by the expression 

- y 2 (abc \H\ hoc) - y 2 (abc \H\ cba), 

the first term corresponding to repulsion of A by B and the second 
to repulsion of A by C. It is reasonable then that the easiest 
path for the reaction would correspond to a linear arrangement 
ABC, the repulsion of A and C then being a minimum for given 
values of tab and r B c Eyring and Polanyi 2 calculated energy 
surfaces for linear configurations by neglecting higher exchange 
integrals and making other simplifying assumptions, the values 

1 A. Farkas, Z. f. phys. Chem. BIO, 419 (1930); P. Harteck and K. H. 
Gbib, ibid. B15, 116 (1931). 

2 H. Eyring and M. Polanyi, Naturwissenschaften 18, 914 (1930); Z. f. 
phy*. Chem. B12, 279 (1931); H. Eyring, Naturwissenschaften 18, 915 
(1930), /. Am. Chem. Soc. 63, 2537 (1931); H. Pelzer and E. Wigner, 
Z.f. phys. Chem. B15, 445 (1932). 



XIV-60] ENERGY OF ACTIVATION OF CHEMICAL REACTIONS 415 

of the Coulomb and single exchange integrals being taken from 
the simple Heitler-London-Sugiura treatment of the hydrogen 
molecule or estimated from the empirical potential function for 
this molecule. These approximate treatments led to values 
in the neighborhood of 10 to 15 kcal for the activation energy. 
Coolidge and James 1 have recently pointed out that the approxi- 
mate agreement with experiment depends on the cancellation of 
large errors arising from the various approximations. 

The similar discussion of the activation energies of a number of 
more complicated reactions has been given by Eyring and 
collaborators. 2 

1 A. S. Coolidge and H. M. James, /. Chem. Phys. 2, 811 (1934). 

2 H. Eyring, /. Am. Chem. Soc. 53, 2537 (1931); G. E. Kimball and 
H. Eyring, ibid. 64, 3876 (1932); A. Sherman and H. Eyring, ibid. 64, 
2661 (1932); R. S. Bear and H. Eyring, ibid. 56, 2020 (1934); H. Eyring, 
A. Sherman, and G. E. Kimball, /. Chem. Phys. 1, 586 (1933); A. Sherman, 
C. E. Sun, and H. Eyring, ibid. 3, 49 (1935). 



CHAPTER XV 

GENERAL THEORY OF QUANTUM MECHANICS 

The branch of quantum mechanics to which we have devoted 
our attention in the preceding chapters, based on the Schrodinger 
wave equation, can be applied in the discussion of most questions 
which arise in physics and chemistry. It is sometimes conven- 
ient, however, to use somewhat different mathematical methods ; 
and, moreover, it has been found that a thoroughly satisfactory 
general theory of quantum mechanics and its physical inter- 
pretation require that a considerable extension of the simple 
theory be made. In the following sections we shall give a brief 
discussion of matrix mechanics (Sec. 51), the properties of angular 
momentum (Sec. 52), the uncertainty principle (Sec. 53), and 
transformation theory (Sec. 54). 

61. MATRIX MECHANICS 

In the first paper written on the quantum mechanics 1 Heisen- 
berg formulated and successfully attacked the problem of calcu- 
lating values of the frequencies and intensities of the spectral 
lines which a system could emit or absorb; that is, of the energy 
levels and the electric-moment integrals which we have been 
discussing. He did not use wave functions and wave equations, 
however, but instead developed a formal mathematical method 
for calculating values of these quantities. The mathematical 
method is one with which most chemists and physicists are not 
familiar (or were not, ten years ago), some of the operations 
involved being surprisingly different from those of ordinary 
algebra. Heisenberg invented the new type of algebra as he 
needed it; it was immediately pointed out by Born and Jordan, 2 
however, that in his new quantum mechanics Heisenberg was 

1 W. Heisenberg, Z. /. Phya. 33, 879 (1925). 
1 M. Born and P. Jordan, ibid. 34, 858 (1925). 

416 



XV-51a] MATRIX MECHANICS 417 

making use of quantities called matrices which had already been 
discussed by mathematicians, and that his newly invented 
operations were those of matrix algebra. The Heisenberg 
quantum mechanics, usually called matrix mechanics, was rapidly 
developed 1 and applied to various problems. 

When Schrodinger discovered his wave mechanics the question 
arose as to the relation between it and matrix mechanics. The 
answer was soon given by Schrodinger 2 and Eckart, 3 who showed 
independently that the two are mathematically equivalent. 

The arguments used by Heisenberg in formulating his quantum 
mechanics are extremely interesting. We shall not present 
them, however, nor enter into an extensive discussion of matrix 
mechanics, but shall give in the following sections a brief treat- 
ment of matrices, matrix algebra, the relation of matrices to wave 
functions, and a few applications of matrix methods to quantum- 
mechanical problems. 

51a. Matrices and Their Relation to Wave Functions. The 
Rules of Matrix Algebra. — Let us consider a set of orthogonal 
wave functions 4 ^ , ^1, • • • , ^n, • • • and a dynamical quantity 



f(qi, pi) 9 the corresponding operator 6 being f ol 



=f (*>5n£) 



In the foregoing chapters we have often made use of integrals 
such as 

fmn = fV&UVndr; (51-1) 

for example, we have given f nn the physical interpretation of the 
average value of the dynamical quantity / when the system is in 
the nth stationary state. Let us now arrange the numbers 
f mn (the values of the integrals) in a square array ordered accord- 
ing to m and n, as follows : 



1 M. Born, W. Heisenberg, and P. Jordan, Z. f. Phys. 36, 557 (1926); 
P. A. M. Dirac, Proc. Roy. Soc. A109, 642 (1925). 

* E. Schrodinger, Ann. d. Phys. 79, 734 (1926). 

• C. Eckart, Phys. Rev. 28, 711 (1926). 

4 These functions include the time factor; a similar discussion can be made 
with use of the functions ^ , 4n 9 • • • not including the time. 

8 In this chapter we shall use the symbol / op . to represent the operator 
corresponding to the dynamical function /. The subscript "op." was not 
used in the earlier chapters because there was no danger of confusion attend- 
ing its omission. See Sees. 10, 12. 



if 00 


At 


/02 


/03 • . - 1 


Ao 


Ai 


/« 


/l3 ... 


/20 


At 


jii 


/23 ... 


J3C 

1. . . 


At 


fzi 


/33 ... 
, 



418 GENERAL THEORY OF QUANTUM MECHANICS [XV-61a 



f = (fmn) = 



This array we may represent by the symbol f or (/ mn ). We 
enclose it in parentheses to distinguish it from a determinant, 
with which it should not be confused. 

We can construct similar arrays g, h, etc. for other dynamical 
quantities. 

It is found that the symbols f, g, h, etc. representing such 
arrays can be manipulated by an algebra closely related to ordi- 
nary algebra, differing from it mainly in the process of multiplica- 
tion. The rules of this algebra can be easily derived from the 
properties of wave functions, which we already know. 

It must be borne in mind that the symbol f does not represent 
a single number. (In particular the array f must not be confused 
with a determinant, which is equal to a single number. There is, 
to be sure, a determinant corresponding to each array, namely, 
the determinant whose elements are those of the array. We have 
set up such determinants in the secular equations of the preceding 
chapters.) The symbol f instead represents many numbers — 
as many as there are elements in the array. The sign of equality 
in the equation f = g means that every element in the array f is 
equal to the corresponding element in the array g. 

Now let us derive some rules of the new algebra. For example, 
the sum of two such arrays is an array each of whose elements 
is the sum of the corresponding elements of the two arrays; that 
is, 

//00 + (700 /oi + 001 /02 +^02 • * *\ 

f + g = ( /10 + 0io /n + 0ii /12 + 012 • • l (51-2) 



It is seen that the arrays add in the same way as ordinary 
algebraic quantities, with f + g = g + f . Addition is com- 
mutative. 

On the other hand, multiplication is not commutative: the product 
fg is not necessarily equal to the product gf. Let us evaluate the 
mnth element of the array fg. It is 

\fg] mn = J^lfop.0op.^ndT. 



XV-51a] MATRIX MECHANICS 419 

Now we can express the quantity g OD & n in terms of the functions 
SFa; with constant coefficients (Sec. 22), obtaining 

k 

That the coefficients are the quantities g kn is seen on multiplying 
by ¥£ and integrating. Introducing this in the integral for 
{fg}mn we obtain 

{fg}mn = %S*ZU.*kdTg kn ; 

k 

since j^^f OJ> ^ k dr is equal to / m *, this becomes 

\fg\mn = ^fmkgicn. (51"3) 

ft 

This is the rule for calculating the elements of the array obtained 
on multiplying two arrays. 

We may continue to develop the algebra of our arrays in this 
way; or we can instead make use of work already done by mathe- 
maticians. The arrays which we have been discussing are called 
matrices y and their properties have been thoroughly investigated 
by mathematicians, who have developed an extensive matrix 
algebra, 1 some parts of which we have just derived. 

Problem 51-1. Show that the laws of ordinary algebra hold for the 
addition and subtraction of matrices and their multiplication by scalars; 
for example, 

f + (g + h) = (f + g) + h, 
at + ag = a(f + g), 
of + M = (a + b)f. 

Matrix methods, especially matrix multiplication, are often 
very useful in solving problems. Thus we have applied Equation 
51-3 in Section 27e, after deriving the equation in order to use it. 
Another example of the use of this equation is provided by 
Problem 51-2. 

In quantum-mechanical discussions the matrix f corresponding 
to the dynamical quantity /(g t -, p») is sometimes defined with the 
use of the wave functions ^ n , which include the time (Eq. 51-1), 
and sometimes with the wave functions ^ w , with the time factor 

1 See, for example, M. B6cher, "Introduction to Higher Algebra," The 
Macmillan Company, New York, 1924. 



420 GENERAL THEORY OF QUANTUM MECHANICS [XV-51a 

omitted, in which case the matrix elements are given by the 
integrals 

fmn = /«/op.Mr. (51-4) 

The matrix elements f mn in the two cases differ only by the time 

2x\(Wm-Wn)t 

factor e h , and as there is no danger of confusion the 
same symbol can be used for the matrix containing the time 
as for that not containing the time. 

Problem 51-2. The elements z mn of the matrix x for the harmonic 
oscillator are given by Equation 11-25. Using the rule for matrix multipli- 
cation, set up the matrices x 2 ( = xx), x 3 , and x 4 , and compare the values of 
the diagonal elements with those found in Section 23a. 

The non-commutative nature of the multiplication of matrices 
is of great importance in matrix mechanics. The difference of 
the product of the matrix q ? - representing the coordinate qj 
and the matrix p, representing the canonically conjugate momen- 

turn pj and the reverse product is not zero, but «— .1, where 1 

Jttz 

is the unit matrix, discussed in the following section; that is, 
these matrices do not commute. On the other hand, q,- and 
p* (with k t* j)j etc., do commute, the complete commutation 
rules for coordinates and momenta being 

h < 
P/q, - q/P, = 2^1, 

p?q* - q*p/ = o, k ?± j,) (5i-5) 

q>q* - q*q/ = o, 
P/P* — p*p? = o. 

These commutation rules together with the rules for converting 
the Hamiltonian equations of motion into matrix form constitute 
matrix mechanics, which is a way of stating the laws of quantum 
mechanics which is entirely different from that which we have 
used in this book, although completely equivalent. The latter 
rules require a discussion of differentiation with respect to a 
matrix, into which we shall not enter. 1 

Problem 51-3. Verify the commutation rules 51-5 by evaluating the 
matrix elements (p,g >),»», etc. 

1 For a discussion of matrix mechanics see, for example, Ruark and Urey, 
"Atoms, Molecules and Quanta," Chap. XVII. 



XV-61b] 



MATRIX MECHANICS 



421 



51b. Diagonal Matrices and Their Physical Interpretation. — A 

diagonal matrix is a matrix whose elements f mn are all zero except 
those with m = n; for example, 



//oo 












/h 














/« 


. . . 


• 








/ks ... 



The unit matrix, 1, is a special kind of diagonal matrix, all the 
diagonal elements being equal to unity: 



1 = 



A constant matrix, a, is equal to a constant times the unit 
matrix: 



1 














1 














1 














1 



a = al = 



a 














a 














a 


. . . 











a . . . 



Application of the rule for matrix multiplication shows that the 
square (or any power) of a diagonal matrix is also a diagonal 
matrix, its diagonal elements being the squares (or other powers) 
of the corresponding elements of the original matrix. 

In Section 10c, in discussing the physical interpretation of the 
wave equation, we saw that our fundamental postulate regarding 
physical interpretation requires a dynamical quantity / to have 
a definite value for a system in the state represented by the wave 
function V n only when f r nn is equal to (/» n ) r , for all values of r. 
We can now express this in terms of matrices: If the dynamical 
quantity f is represented by a diagonal matrix f then this dynamical 
quantity has the definite value f nn for the state corresponding to the 
wave function V n of the set ¥ , ^i, • • • . 

For illustration, let us discuss some of the wave functions which 
we have met in previous chapters. The solutions 




422 GENERAL THEORY OF QUANTUM MECHANICS [XV-51b 

/ 2xiWot \ 

*o\ = h* h ),*i, ' ' ' 
of the wave equation for any system correspond to a diagonal 
energy matrix 

Wo 

-0- JFi— 0— 

H = W 2 
W 

so that, as mentioned in Section 10c, the system in a physical 
condition represented by one of these wave functions has a 
fixed value of the total energy. 

In the case of a system with one degree of freedom no other 
dynamical quantity (except functions of H only, such as H 2 ) 
is represented by a diagonal matrix; with more degrees of freedom 
there are other diagonal matrices. For example, the surface- 
harmonic wave functions Qi m (&)® m (<p) for the hydrogen atom 
and other two-particle systems separated in polar coordinates 
(Sees. 19, 21) make the matrices for the square of the total 
angular momentum and the component of angular momentum 
along the z axis diagonal, these dynamical quantities thus having 
definite values for these wave functions. The properties of 
angular momentum matrices are discussed in Section 52. 

The dynamical quantities corresponding to diagonal matrices 
relative to the stationary-state wave functions SP'o, ^i, • • • are 
sometimes called constants of the motion of the system. The 
corresponding constants of the motion of a system in classical 
mechanics are the constants of integration of the classical equa- 
tion of motion. 

Let us now consider a system whose Schrodinger time functions 
corresponding to the stationary states of the system are ^ , 
^i, • • • , ^n, • • • . Suppose that we carry out an experiment 
(the measurement of the values of some dynamical quantities) 
such as to determine the wave function uniquely. Such an 
experiment is called a maximal measurement. A maximal 
measurement for a system with one degree of freedom, such as 
the one-dimensional harmonic oscillator, might consist in the 
accurate measurement of the energy; the result of the measure- 
ment would be one of the characteristic energy values W n \ and 
the corresponding wave function ^ n would then represent the 



XV-51b] MATRIX MECHANICS 423 

system so long as it remain undisturbed and could be used for 
predicting average values for later measurements (Sees. 10, 12;. 
A maximal measurement for a system of three degrees of freedom, 
such as the three-dimensional isotropic harmonic oscillator or 
the hydrogen atom with fixed nucleus and without spin, might 
consist in the accurate determination of the energy, the square 
of the total angular momentum, and the component of the 
angular momentum along the z axis. The wave function cor- 
responding to such a maximal measurement would be one of 
those obtained by separating the wave equation in polar coordi- 
nates, as was done in Chapter V. 

It is found that the accurate measurement of the values of N 
independent 1 dynamical quantities constitutes a maximal 
measurement for a system with N degrees of freedom. In 
classical mechanics a maximal measurement involves the accurate 
determination of the values of 2N dynamical quantities, such as 
the N coordinates and the N momenta, or for a one-dimensional 
system the energy and the coordinate, etc. A discussion of the 
significance of this fact will be given in connection with the 
uncertainty principle in Section 53. 

Now let us consider a complete set of orthogonal normalized 
wave functions xo, Xi> * * * > Xn', • • • , each function Xn' being 
a solution of the Schrodinger time equation for the system under 
discussion. These wave functions are linear combinations of 
the stationary-state wave functions ^ n , being obtained from 
them by the linear transformation 

Av = ^a n > n * n , (51-6) 

n 

in which the coefficients a n > n are constants restricted only in 
that they are to make the x's mutually orthogonal and normal- 
ized. A set of wave functions Xn' is said to form a representation 
of the system. Corresponding to each representation matrices 2 
f, g', etc. can be constructed for the dynamical quantities 
/, g, etc., the elements being calculated by equations such as 

fm>n> = JXm'/op.Xn'dT (51~7) 

1 The meaning of independent will be discussed later in this section. 

2 We use primed symbols to indicate that the matrices correspond to the 
representation xn f - 



424 GENERAL THEORY OF QUANTUM MECHANICS [XV-61b 

or obtained from the matrices f, g, etc. (corresponding to the 
stationary-state representation ^ n ), by the use of the coefficients 
a n > n of Equation 51-6. 

So far we have discussed the measurement of constants of the 
motion of the system only; that is, of quantities which are repre- 
sented by diagonal matrices relative to the Schrodinger wave 
functions >Po, ^\, ^2, • • • > and which are hence independent of 
the time. But in general we might make a maximal measure- 
ment consisting in the accurate measurement of N dynamical 
quantities /, g*, etc., whose matrices f, g, etc., relative to ^ , 
#1, • • • , are not all diagonal matrices. In the case of such a 
maximal measurement we must specify the time t = t f at which 
the measurement is made. An accurate measurement of the 
quantities /, g, etc. at the time t = t' requires that at the time 
t = t' the matrices f, g', etc. be diagonal matrices. In order to 
find the wave function representing the system at times subse- 
quent to t = t' (so .long as the system remain undisturbed), 
we must find the representation Xn* which makes these matrices 
diagonal at the time t = t'. The accurate values of /, g f etc. 
obtained by measurement will be identical with the numbers 
fn'n'y g n 'n', etc., occurring as a certain diagonal element of the 
diagonal matrices f , g', etc., and the wave function representing 
the system will be the corresponding Xn'. 

It is interesting to notice that the condition that the dynamical 
quantity / be represented by a diagonal matrix f ' in the x repre- 
sentation can be expressed as a differential equation. In order 
for V to be a diagonal matrix, /4v must equal for w! not equal 
to n' and a constant value, /„'«', say, for m! = n'. This means 
that on expanding / op .Xn' in terms of the complete set of functions 
X only the one term fn'n'Xw will occur; that is, that 

/op.Xn' = /n'«'Xn', (51-8) 

in which f n ' n * is a number, the w'th diagonal element of the 
diagonal matrix f. For example, the stationary-state wave 
functions V n im for a hydrogen atom as given in Chapter V 
satisfy three differential equations, 



HopVnlm = Wr&nlm, 

1(1 + l)h* 
4ir 2 






XV-62] THE PROPERTIES OF ANGULAR MOMENTUM 425 

and 

TUf yT, _ m ^ l ^rr 

-ft* z op. *nZm — "q~ n ^ m > 

corresponding to the three dynamical functions whose matrices 
are diagonal in this representation ; namely, the energy, the square 
of the total angular momentum, and the z component of the 
angular momentum. For a discussion of this question from a 
different viewpoint see the next section. 

62. THE PROPERTIES OF ANGULAR MOMENTUM 

As pointed out in the previous section, systems whose wave 
equations separate in spherical polar coordinates (such as the 
hydrogen atom) possess wave functions corresponding not only 
to definite values of the energy but also to definite values of the 
total angular momentum and the component of angular momen- 
tum along a given axis (say, the z axis). In order to prove this 
for one particle 1 let us construct the operators corresponding 
to M x , My, and M z , the components of angular momentum along 
the x, y, and z axes. Since classically 

Mx = VPz - ZVy, (52~1) 

with similar expressions for M v and M z , the methods of Section 
10c for constructing the operator corresponding to any physical 
quantity yield the expressions 

A, 
= 2xA 



M x < 

My 

M t0 



2wi\ dx dz/[ 

( ±- ±\ 

\ X dy y dx)' 



2<7tt 



In order to calculate the average values of these quantities it is 
convenient to express them in terms of polar coordinates. By the 
standard methods (see Sec. 16) we obtain 



k**(±iA-cot*£) (52-3) 



■L'J- X op. IE I'lVi. y op. — O *' 

1 The total angular momentum and its z component also have definite 
values for a system of n particles in field-free space; see, for example, Born 
and Jordan, "Elementare Quantenmechanik, ,, Chap. IV, Julius Springer, 
Berlin, 1930. 



426 GENERAL THEORY OF QUANTUM MECHANICS [XV-52 
and 

M ~ - ss £ (52 " 4) 

We have postulated that the wave equation is separable in 
polar coordinates; if we also restrict the potential energy to be a 
function of r alone, the dependence of the wave functions on the 
angles will be given by 

hm n (&, v, r) = e^Mri^iW, (52-5) 

where 0jm(#)<£m(^>) are the surface-harmonic wave functions 
obtained in Section 18. Using these and the expressions in 
Equations 52-3 and 52-4, we can evaluate the integrals of the 
type 

M x (l'm'; Im) = fy?'m>nM zv di mn dT. (52-6) 

In order to prove that the square of the total angular momentum 
M 2 has a definite value for a given stationary state described by 
4>i m n, it is necessary for us to show that the average value of any 
power of M 2 is identical with the same power of the average value 
(Sec. 10c). By using the properties of matrices given in the 
previous section we can considerably simplify this proof. As 
stated there, we need only show that Af 2 D . is represented by a 
diagonal matrix. Furthermore we can obtain the matrix 
for M 2 from the matrices for M x , M VJ and M & by using the relation 
defining M 2 in classical mechanics, 

M 2 = Ml + M 2 + M 2 } (52-7) 

and applying the rules for matrix multiplication and addition. 

If we carry out this procedure, we first find on evaluation of 
the proper integrals that 

MJ}'m'\ Im) = -^[W + 1) - m(m + l)}^ m ,, m+1 + 

{1(1 + 1) - m(m - l))»8 m >, m ~i]8i> th (52-8) 
ih 

My(Vm') Im) = ^[{1(1 + 1) - m(m + l)}^, m+1 - 

{1(1 + 1) - m(m - l)}^,^]^,*, (52-9) 
M,(l'm'; Im) = ~m5 r ,j3 m ', m , (52-19) 

in which 5 m ', m+ i = 1 for m' = m + 1 and otherwise, etc. 



XV-52] THE PROPERTIES OF ANGULAR MOMENTUM 427 

The next step is to obtain the elements of the matrices M*, 
M 2 , and M 2 from these by using matrix multiplication, and then 
the elements of M 2 by using matrix addition. The final result 

is that M 2 is a diagonal matrix with diagonal elements . 2 

It is therefore true that M 2 has a definite value in the state 
ypimn] in other words, it is a constant of the motion with the 

value «<+££ 

47T 2 

The proof that M z is also a constant of the motion is contained 
in Equation 52-10, which shows that M* is a diagonal matrix 
with diagonal elements mh/2w so that its value is mh/2w for the 
state with quantum number m. 

Problem 62-1. Carry out the transformation of Equations 52-2 into 
polar coordinates. 

Problem 52-2. Derive Equations 52-8, 52-9, and 52-10. 

Problem 52-3. Obtain the matrices for Ml, M\, M\ by matrix multi- 
plication and from them obtain the matrix for M 2 . 

There is a close connection between the coordinate system in 
which a given wave equation is separated and the dynamical 
quantities which are the constants of the motion for the resulting 
wave functions. Thus for a single particle in a spherically 
symmetric field the factor S(d, <p) of the wave function which 
depends only on the angles satisfies the equation (see Sec. 18a) 

4^ //sin fl^f) + -J— d ^ 2 = -1(1 + 1)5. (52-11) 

sin & d&\ ddj sin 2 & dip 2 v ■ / v 

It can be shown that the operator for M 2 in polar coordinates 
has the form 

Ml. = -^j-4-5 A( sin »Si) + -^1-i -Til' < 52 ~ 12 ) 
p 47r 2 (sin # d&\ d$/ sin 2 # dip 2 ) 

so that Equation 52-11 may be written 

ttu = 1(1 + l)^+ni m , (52-13) 

since \l/ = S(&, <p)R(r) and M 2 V . does not affect R(r). 

Furthermore the equation for $,»(<?), the <p part of ^, is (Sec. 
18a) 

g = _ m ** f (52-14) 



428 GENERAL THEORY OF QUANTUM MECHANICS [XV-58 
whereas from Equation 52-4 we find that 

M -- = -SS' (52_15) 

so that Equation 52-13 may be written in the form 

M? ov .+ nlm = m*£&ni». (52-16) 

The formal similarity of Equations 52-13 and 52-16 with the 
wave equation 

is quite evident. All three equations consist of an operator 
acting upon the wave function equated with the wave function 
multiplied by the quantized value of the physical quantity repre- 
sented by the operator. Furthermore, the operators jf/ p., 
M%^, and AfJ op . will be found to commute with each other; 
that is, 

H ov .{Mlx) = MlXH ov . x ), 

etc., where x is any function of #, <p } and r. 

It is beyond the scope of this book to discuss this question more 
thoroughly, but the considerations which we have given above for 
this special case can be generalized to other systems and other 
sets of coordinates. Whenever the wave equation can be 
separated it will be found that the separated parts can be thrown 
into the form discussed above, involving the operators of several 
physical quantities. These physical quantities will be constants 
of the motion for the resulting wave functions, and their operators 
will commute with each other. 

63. THE UNCERTAINTY PRINCIPLE 

The Heisenberg uncertainty principle 1 may be stated in the 
following way: 

The values of two dynamical quantities f and g of a system can be 
accurately measured at the same time only if their commutator is 
zero; otherwise these measurements can be made only with an 
uncertainty AfAg whose magnitude is dependent on the value of the 
commutator. In particular, for a canonically conjugate coordinate 

1 W. Hbisknbbrg, Z. f. Phys. 43, 172 (1927). 



XV-63] THE UNCERTAINTY PRINCIPLE 429 

q and momentum p the uncertainty AgAp is of the order of rruignitude 
of Planck's constant h, as is AWAJ for the energy and time. 

To prove the first part of this principle, we investigate the 
conditions under which two dynamical quantities / and g can be 
simultaneously represented by diagonal matrices. Let these 
matrices be f and g', Xn' being the corresponding representation. 
The product f'g' of these two diagonal matrices is found on 
evaluation to be itself a diagonal matrix, its n'th element being 
the product of the n'th diagonal elements f n > and g n > of the 
diagonal matrices f and g'. Similarly g'f is a diagonal matrix, 
its diagonal elements being identical with those of f'g'. Hence 
the commutator of f and g' vanishes: f'g' — g'f = 0. The 
value of the right side of this equation remains zero for any 
transformation of the set of wave functions, and consequently 
the commutator of f and g vanishes for any set of wave functions; 
it is invariant to all linear orthogonal transformations. We 
accordingly state that, in order for two dynamical quantities f and g 
of a system to be accurately measurable at the same time, their com- 
mutator must vanish; that is } the equation 

fg - gf = (53-1) 

nust hold. 

A proof of the second part of the uncertainty principle is 
lifficult; indeed, the statement itself is vague (the exact meaning 
of A/, etc., not being given). We shall content ourselves with 
the discussion of a simple case which lends itself to exact treat- 
ment, namely, the translational motion in one dimension of a 
free particle. 

The wave functions for a free particle with coordinate x are 

2iriV2mW(x-xo) 2riWt 

Ne h e h (Sec. 13), the positive sign in the first 

exponential corresponding to motion in the x direction and the 
negative sign in the — x direction. On replacing W by p|/2m 

2iripx(x — Xg) 2«tpj< 

this expression becomes Ne h e 2mh , in which positive 
and negative values of the momentum p z refer to motion in the x 
direction and the —x direction, respectively. A single wave 
function of this type corresponds to the physical condition in 
which the momentum and the energy are exactly known, that 
is, to a stationary state of the system. We have then no knowl- 
edge of the position of the particle, the uncertainty Ax in the 



430 GENERAL THEORY OF QUANTUM MECHANICS [XV-53 

coordinate x being infinite, as is seen from the probability dis- 
tribution function ty*ty, which is constant for all values of x 
between — c© and + oo . When Ap x is zero Ax is infinite. 

Now let us suppose that at the time I = we measure the 
momentum p x and the coordinate x simultaneously, obtaining 
the values p and x , with the uncertainties Ap x and Ax, respec- 
tively. Our problem is to set up a wave function x which 
represents this physical condition of the system One way 
of doing this is the following. The wave function 

/°° (Px— Po) a 2vipx(x~xo) 2irivlt 

j 2(Ap,)> e h e 2mk ( ip x (53-2) 

(Pr-P0)» 

corresponds to a Gaussian-error-curve distribution e (A/>r) ' 
of the values of the momentum p x about the average value p , 
with the uncertainty 1 Ap x . (The factor Yi in the exponent in 
Equation 53-2 results from the fact that the coefficients of the 
wave functions are to be squared to obtain probability values.) 
A is a normalization constant. On evaluating the integral we 
find for x at the time t = the expression 

_ 27r2(Ap r)H*-so) g , 2ytp (3g-jo) 

x(0) = Be h> + h 9 (53_ 3) 

which corresponds to the probability distribution function 
for x 

X*(0)x(0) = BH W (53-4) 

with 

** - sfe (53 - 5) 

This is also a Gaussian error function, with its maximum at 
x = x and with uncertainty Ax given by Equation 53-5. It is 
seen that the wave function x corresponds to the value h/2ir 
for the product of the uncertainties Ax and Ap x at the time t = 0, 
this value being of the order of magnitude h, as stated at the 
beginning of the section. 

Problem 53-1. Evaluate the normalization constants A and B 2 by 
carrying out the integration over p x and then over x. 

1 The quantity Ap x is the reciprocal of the so-called precision index of 
the Gaussian error curve and is larger than the probable error by the factor 
2.10; see R. T. Birge, Phys. Rev. 40, 207 (1932). 



XV-53] THE UNCERTAINTY PRINCIPLE 431 

Problem 63-2. Carry through the above treatment, retaining the time 
factors. Show that the center of the wave packet moves with velocity 
po/m, and that the wave packet becomes more diffuse with the passage of 
time. 

A gener&l discussion* by the use of the methods of transforma- 
tion theory (Sec. 54), which we shall not reproduce, leads to the 
conclusion that the product of the uncertainties AfAg accompany- 
ing the simultaneous measurement of two dynamical quantities 
/ and g is at least of the order of magnitude of the absolute value 
of the corresponding diagonal element in their commutator 

fg — gf» (The commutator of x and p z is ^— .1 (Eq. 51-5), the 

absolute value of the diagonal elements being A/27T, in agreement 
with the foregoing discussion.) This leads to the conclusion that 
the energy W and time t are related regarding accuracy of measure- 
ment in the same way as a coordinate and the conjugate momen- 
tum, the product of the uncertainties AW and At being of the 
order of magnitude of h (or h/2ir) . In order to measure the energy 
of a system with accuracy AW, the measurement must be 
extended over a period of time of order of magnitude h/AW. 

Problem 53-3. Show that the commutator Wt - tW has the value 

h h A 

— — — .1 by evaluating matrix elements, recalling that W ov — and 

^Trt ' 2iri dt 

top. == *. 

It is natural for us to inquire into the significance of the 
uncertainty principle by analyzing an experiment designed to 
measure x and p x . Many "thought experiments" have been 
discussed in the effort to find a contradiction or to clarify the 
theory; in every case these have led to results similar to the 
following. Suppose that we send a beam of light of frequency 
v along the axis AO of Figure 53-1, and observe along the 
direction OB to see whether or not the particle, restricted to 
motion along the x axis, is at the point or not. If a light 
quantum is scattered into our microscope at B, we know that 
the particle is in the neighborhood of 0, and by analyzing the 
scattered light by a spectroscope to determine its frequency v f 
we can calculate the momentum of the particle by use of the 
equations of the Compton effect. But for light of finite fre- 
quency the resolving power of the microscope is limited, and oui 



432 GENERAL THEORY OF QUANTUM MECHANICS [XV-54 

measurement of x will show a corresponding uncertainty Ax, 
which decreases as the frequency increases. Similarly the 
measurement of the momentum by the Compton effect will show 
an uncertainty Ap x , increasing as the frequency increases. 
The detailed analysis of the experiment shows that under the 
most favorable conditions imaginable the product AxAp* is of 
the order of magnitude of h. l 




/ 






i / 

!/ 



i 

i 



hV 



Source of light 
Fia. 53-1. — Diagram of experiment for measuring x and p x of particle. 

64. TRANSFORMATION THEORY 

In discussing the behavior of a system the following question 
might arise. If at the time t = t' the dynamical property / is 



1 For the further discussion of the uncertainty principle see W. Heisenberg, 
"The Physical Principles of the Quantum Theory," University of Chicago 
Press, Chicago, 1930; N. Bohr, Nature 121, 580 (1928); C. G. Darwin, Proc. 
Roy. Soc. A117, 258 (1927); A. E Ruark, Phys. Rev. 31, 311, 709 (1928); 
E. H. Kennard, Phys. Rev. 31, 344 (1928); H. P. Robertson, Phys. Rev. 34, 
163 (1929); 35, 667.(1930); 46, 794 (1934); and also Ruark and Urey, "Atoms, 
Molecules and Quanta," Chap. XVIII; and other references listed at the 
end of the chapter. 



XV-M] TRANSFORMATION THEORY 433 

found on measurement to have the value /', what is the prob- 
ability that the immediately subsequent measurement of the 
dynamical property g will yield the value g"! We know one 
way to answer this question, namely, to find the wave function 1 
X (one of the representation which makes V a diagonal matrix) 
corresponding to the value /' of /, to use it to calculate the 
average value of all powers of g, and from these to construct a 
probability distribution function for g. This is not a very 
simple or direct procedure, however; it is of interest that an 
alternative method has been found by means of which these 
probability distribution functions can be calculated directly. 
This method, called the transformation theory* is a general quan- 
tum mechanics within which wave mechanics is included, the 
Schrodinger wave equation being one of a large number of 
equations of the theory and the Schrodinger wave functions a 
particular type of transformation functions. We shall not 
enter into an extensive discussion of transformation theory but 
shall give only a brief description of it. 

Let us represent by ((/'I/') a probability amplitude function or 
transformation function such that (g'\f')*(g'\f) is the probability 
under discussion, (g'\f)* being the complex conjugate of (g'\f). 
[In case that g' can be any one of a continuum of values, 
(g'\f')*(g'\f) is interpreted as a probability distribution function, 
the probability that g have a value between g r and g f + dg' 
being (g'\f')*(9'\f'W.] 

The Schrodinger stationary-state wave functions are proba- 
bility amplitude functions between the energy and the coordi- 
nates of the system. For a system with one degree of freedom, 
such as a harmonic oscillator, the wave functions \p n are the 
transformation functions (x'\W f ) between the coordinate x and 
the characteristic energy values, and for the hydrogen atom 
the wave functions \l/ n im(r, #, <p), discussed in Chapter V, are the 
transformation functions (r^VVto) between the coordinates 
r, #, and <p of the electron relative to the nucleus and the charac- 
teristic energy values W n , the square of angular momentum values 

1 In case that the measurement of / is not a maximal measurement many 
wave functions might have to be considered. 

* The transformation theory was developed mainly by P. A. M. Dirac, 
Proc. Roy. Soc. A113. 621 (1927). and P. Jordan, Z.f. Phys. 40, 809 (1927)} 
44. 1 (1927). 



434 GENERAL THEORY OF QUANTUM MECHANICS [XV-54 

— j— 2 — ) and the angular momentum component values mh/2Tr 9 

represented by the symbols n, l, and m, respectively. 

Two important properties of transformation functions are the 
following: 

The transformation function between / and g is equal to that 
between g and /: 

(fV) = (g'\n* (54-D 

The transformation function between / and h is related to that 
between / and g and that between g and h by the equation 

(f'W) = f(f'\9')*(9'\h'W. (54-2) 

In this equation the integration includes all possible values g' 
which can be obtained by measurement of g; in case that g f 
represents a set of discrete values, the sum over these is to be 
taken. 

We have often Written the Schrodinger wave equation in the 
form 

In the nomenclature of transformation theory this is 

W f representing a characteristic value W n of the energy and 
(<7;l^') the corresponding transformation function to the coordi- 
nates g,-. In transformation theory it is postulated that a similar 
equation 

*.,„(/¥) = qV'W) (54-3) 

is satisfied by every transformation function (/%')• In this 
equation g ov . is the operator in the / scheme representing 
the dynamical quantity g. We shall not discuss the methods 
by means of which the / scheme of operators is found but shall 
restrict our attention to the q scheme, in which the operators 

are obtained by the familiar method of replacing pk by *—. - — 

Zti ogic 

The transformation functions are normalized and mutually 

orthogonal, satisfying the equation 

Kf'lffWlf'W = h fr . (54-4) 



XV-64] .. TRANSFORMATION THEORY 435 

It is interesting to note that this equation signifies that, if the 
dynamical quantity / has been found on measurement to have 
the value /', immediate repetition of the measurement will give 
the same value/' with probability unity, inasmuch as the integral 
of Equation 54-4 is the transformation function (/'I/") (see 
Eq. 54-2) and Equation 54-4 requires it to vanish except when 
/" is equal to/', in which case it has the value 1. 

From the above equations we can find any transformation 
function (/'|#'), using the q system of operators only, in the 
following way: we find the transformation functions (q'\f) and 
WW) by solving the corresponding differential equations 
54-3, and then obtain (/%') by integrating over the coordi- 
nates (Eq. 54-4). As an example, let us obtain the transforma- 
tion function 0^1^') between the energy W and the linear 
momentum p x of a one-dimensional system. The function 
Or' | IT') is the Schrodinger wave function, obtained by solving 
the wave equation 

H^(x'\W) = W'(z'\W) 

as described in the preceding chapters of the book. The trans- 
formation function (x'\p' x ) between a Cartesian coordinate and 
its canonically conjugate momentum is the solution of the 
equation 

P^(*'lPi) = v' x {Av' x ) 

or 

and hence is the function 

2tjx' P ' x 

(x'\ V ' x ) = Ce » , (54-5) 

C being a normalizing factor. The transformation function 
(p' x \W), the momentum probability amplitude function for a 
stationary state of the system, is accordingly given by the 
equation 

2tjx'pZ 

(P',\W) = JCe h (x'\W'W (54-6) 

or 

2idx'p' x 

(P' 9 \W») = fCe * M*'W- (54-7) 



436 GENERAL THEORY OF QUANTUM MECHANICS [XV-64 

On application of this equation it is found that the momentum 
wave functions for the harmonic oscillator have the same form 
(Hermite orthogonal functions) as the coordinate wave functions 
(Prob. 54-1), whereas those for the hydrogen atom are quite 
different. 1 

Problem 54-1. Evaluate the momentum wave functions for the harmonic 
oscillator. Show that the average value of p r x for the nth state given by 
the equation 



/: 



t (p.\w»)*(p' M \w,) P ';dp'. 

is the same as given by the equation 



I>fe)'£- 



Problem 64-2. Evaluate the momentum wave function for the normal 
hydrogen atom, 

(p x P v Vz\^m) = J J JCe h (x'y'z'\nlm)dx'dy'dz'. 

It is convenient to change to polar coordinates in momentum space as well 
as in coordinate space. 

The further developments of quantum mechanics, including the 
discussion of maximal measurements consisting not of the 
accurate determination of the values of a minimum number of 
independent dynamical functions but of the approximate meas- 
urement of a larger number, the use of the theory of groups, the 
formulation of a relativistically invariant theory, the quantiza- 
tion of the electromagnetic field, etc., are beyond the scope of 
this book. 

General References on Quantum Mechanics 

Matrix mechanics: 

M. Born and P. Jordan: "Elementare Quantenmechanik," Julius 
Springer, Berlin, 1930. 

Transformation theory and general quantum mechanics: 

P. A. M. Dirac: ''Quantum Mechanics," Oxford University Press, New 
York, 1936. 

^he hydrogen-atom momentum wave functions are discussed by B. 
Podolsky and L. Pauling, Phys. Rev. 34, 109 (1929), and by E. A. Hylleraas, 
Z./.P/iy«.74,216(1932). 



XV-54] TRANSFORMATION THEORY 437 

J. v. Neumann: " Mathematische Grundlagen der Quantenmechanik," 
Julius Springer, Berlin, 1932. 

Questions of physical interpretation: 

W. Heisenbbrg: "The Physical Principles of the Quantum Theory," 
University of Chicago Press, Chicago, 1930. 

General references: 

A. E. Rtjark and H. C. Urey: "Atoms, Molecules and Quanta," 
McGraw-Hill Book Company, Inc., New York, 1930. 

E. U. Condon and P. M. Morse: "Quantum Mechanics," McGraw-Hill 
Book Company, Inc., New York, 1929. 

A. Sommerfeld: "Wave Mechanics," Methuen & Company, Ltd., 
London, 1930. 

H. Weyl: "The Theory of Groups and Quantum Mechanics," E. P. 
Dutton & Co., Inc., New York, 1931. 

J. Frenkel: "Wave Mechanics," Oxford University Press, New York. 
1933. 



APPENDIX I 

VALUES OF PHYSICAL CONSTANTS 1 

Velocity of light c = 2.99796 X 10 10 cm sec" 1 

Electronic charge e = 4.770 X 10~ 10 abs. e.s.u. 

Electronic mass w = 9 . 035 X 10~ M g 

Planck's constant h = 6 . 547 X 10~ 27 erg sec 

Avogadro's number N = 0.6064 X 10 24 mole" 1 

Boltzmann's constant k = 1 . 3709 X 10~ 16 erg deg" 1 

2*-e 2 

Fine-structure constant a — = 7.284 X 10""* 

he 

Radius of Bohr orbit in normal hydro- 
gen, referred to center of mass a = 0.5282 X 10~ 8 cm 

Rydberg constant for hydrogen Zfe = 109677.759 cm -1 

Rydberg constant for helium # He = 109722 . 403 cm -1 

Rydberg constant for infinite mass R„ = 109737.42 cm -1 

h 
Bohr unit of angular momentum — = 1 . 0420 X 10"" 27 erg sec 

Magnetic moment of 1 Bohr magneton /*o = 0.9175 X 10"" 20 erg gauss"" 1 

Relations among Energy Quantities 

1 erg - 0.6285 X 10 12 v.e. = 0.5095 X 10 16 cm" 1 = 1.440 X 10 18 cal/mole 
1.591 X 10" 12 erg = 1 v.e. = 8106 cm" 1 = 23055 cal/mole 
1.963 X 10" 18 erg = 1.234 X 10~ 4 v.e. = 1 cm"" 1 = 2.844 cal/mole 
0.6901 X 10- 18 erg = 4.338 X 10" 6 v.e. = 0.3516 cm" 1 = 1 cal/mole 

1 These values are taken from the compilation of R. T. Birge, Rev. Mod. 
Phys. 1, 1 (1929), *as recommended by Birge, Phys. Rev. 40, 228 (1932). 
For probable errors see these references. 



439 



APPENDIX II 

PROOF THAT THE ORBIT OF A PARTICLE MOVING 
IN A CENTRAL FIELD LIES IN A PLANE 

The force acting on the particle at any instant is in the direc- 
tion of the attracting center (see F, Fig. 1). Let the arrow 
marked v in the figure represent the direction of the motion at 
any instant. Set up a system of Cartesian axes xy z with origin 
at the point P and oriented so that the z axis points along v and 
the y axis points perpendicular to the plane of F and v, being 
directed up from the plane of the paper in the figure. 

z 






y 

Fio. II-l. 
Then the equation of motion (in Newton's form) in y is 

since there is no component of the force F in the y direction. 
Therefore the acceleration in the y direction is zero and the 
velocity in the y direction, being initially zero, will remain zero, 
30 that the particle will have no tendency to move out of the plane 
ietermined by F and v- 



440 



APPENDIX III 

PROOF OF ORTHOGONALITY OF WAVE FUNCTIONS 
CORRESPONDING TO DIFFERENT ENERGY LEVELS 

We shall prove that, if W n ^ Wk, the solution ^„ of the 
wave equation 

N 



2^» + TprV. - V)*. = (l) 

and the solution \p£ of the equation 

N 
t-1 

satisfy the relation 

mtndr = 0; (3) 

i.e., that ypk is orthogonal to \p n . 

Multiply Equation 1 by \f/*, Equation 2 by ^ n . and subtract 
the second from the first. Since V is real, the result is that 

N 

2^***"*" - *•*#*> + 7? (Tf " - *W*» = o. (4) 

t-1 

If we now integrate the terms of this equation over configuration 
space, we obtain 

N 

~(W % - Wu) fitter = -2^J ( ** V ' V,i " **&*)*• (5) 

t-1 

If we introduce the expression for v» ? in terms of Cartesian coordi- 
nates into the integral on the right, it becomes 

J-l 

441 



442 APPENDIX III 

in which we have written qi, g 2 ? • • • , Qzn in place of x i} y Xi Zi t 
Xi, - - - > «at. We next make use of the identity 






'd qi 
from which we see that 

because of the boundary conditions on ^. 

Since every term of the sum can be treated similarly, the 
expression 6 is equal to zero and therefore 



^V» - Wkij+t+ndr = 0, 



from which Equation 3 follows, since W n — Wk ^ U. 

If Tf n s Jf*, so that ^fc and ^„ are two linearly independent 
wave functions belonging to the same energy level, \p k and \p n are 
not necessarily orthogonal, but it is always possible to construct 
two wave functions i/^ and \f/' n > belonging to this level which are 
mutually orthogonal. This can be done in an infinite number of 
ways by forming the combinations 

Vk> = c&k + Mn and <A«' = a'+k + 0tyn, 

with coefficients a, p, a', ft satisfying the relation 

WWn'dr = a*a'S+t+ k dT + a*ft m+ndr + a'pfri+dr + 

P*P'm+jT = 0. (8) 



APPENDIX IV 

ORTHOGONAL CURVILINEAR COORDINATE SYSTEMS 

In Section 16 the general formulas for the Laplace operator v 2 
and for the volume element dr were given in terms of the quanti- 
ties q u , q vy and q w defined by Equation 16-4. In this appendix 
there are given the equations of transformation (in terms of 
Cartesian coordinates) and the expressions for the q'a for the 11 
sets of orthogonal coordinate systems listed by Eisenhart 1 as the 
only such systems in which the three-dimensional Schrodinger 
wave equation can be separable. In addition the explicit 
expressions 2 for v 2 and dr are given for a few of the more impor- 
tant systems. These quantities may be obtained for the other 
systems by the use of Equations 16-3 and 16-5. 

Cylindrical Polar Coordinates 

X = p COS <p, 

y = p sin <p, 

z = z. 

q P = 1, q a = 1, q<p = p. 
dr = pdpdzdip. 

v p d P \%J "■" p 2 d<p 2 "*■ dz 2 
Spherical Polar Coordinates 

x = r sin # cos <p, 
y = r sin & sin <p, (Fig. 1-1), 
z = r cos #. 
q r = 1, ^ = r, <k = r sin #. 
dr = r 2 sin &drd&d<p. 



v2 = T 2 l( r fr) + ra^ l>( 8in * £) 



+ 



1 



r 2 sin 2 # d<p 2 



1 L. P. Eisenhart, Phys. Rev. 45, 428 (1934). 

2 E. P Adams, "Smithsonian Mathematical Formulae," Washington, 
1922. This book contains extensive material on curvilinear coordinates as 
well as other very useful formulas. 

443 



444 APPENDIX IV 

Parabolic Coordinates 

X = \/^7 COS (p, 

y = Vfr sin <p, 

«/€ = 2\ — F""' ft = 2\"~V~ ; ft = vw- 

d T = /4(£ + y)d£dT)d<p. 

2 = _4__ _*/ 1*^ , _i_ 1/ l\ , 1 i! 

v £ + v a*\ w * + * M%/ fr a^ 2 ' 

Confocal Elliptic Coordinates (Prolate Spheroids) 

x = as/^ 2 — 1\/1 — *7 2 cos v?, 
2/ = a-\/£ 2 — 1a/1 ~~ *? 2 sin <p, 
z = a^. 

In terms of the distances r A and r B from the points (0, 0, — a) 
and (0, 0, a), respectively, £ and rj are given by the expressions 

t _ r A + r B r A - r B 

5 ~ 2a ; ^ 2a 

«« = a V5^r' * = Wf=?' ft = aV( * 2 - Da - r? 2 ). 

dr = a 3 (£ 2 - r} 2 )d&r)d<p. 

} 



+ 

e - v* 



(e - D(i - r, 2 ) d*>*J 

Spheroidal Coordinates (Oblate Spheroids) 
x = a£j cos ip, y = a£»> sin ?,« = a\/(£ 2 - 1)(1 - v 2 ). 

ft - a \ ^2 Z i ' 9- = a -\j l H v 2 > 9* = aft- 

Parabolic Cylinder Coordinates 

x = %(u - v), y - Vuv, z =2. 

. 1 ju + v n 1 \ u + v 
? " = 2\T"' qv = ^V - ^' 9 * = L 



APPENDIX IV 445 

Elliptic Cylinder Coordinates 



x = ay/(u 2 — 1)(1 — v 2 ), y = auv, z = z. 

\u 2 - v 2 \u 2 - V 2 

Qu = a yl u 2 - i ' Qv = a \ i - v * ' q ° = 

Ellipsoidal Coordinates 

2 _ (a 2 +^)( a 2 + t; )(a2 + ^) 2 _ (b* + u)(b* + v)(b* + w) 
X (a 2 - 6 2 )(a 2 - c 2 ) ' y (6 2 - c 2 )(b 2 - a 2 ) ' 



., = (c 2 + u)(c 2 + v)(c 2 + w) 
(c 2 - a 2 )(c 2 - b 2 ) 



*2 = 



2 = (u - v)(u - w) 2 (v - ti?)(ty - u) 

qu 4(a 2 + u)(b 2 + u)(c 2 + u)' q * 4(a 2 + v)(b 2 + v)(c 2 + v)' 



ft = 



(w — w)(w — v) 



4(a 2 + w)(6 2 + w)(c 2 + w) 
Confocal Parabolic Coordinates 

x = ^{u + v + w - a - 6), y 2 = fr - fl / 

t (6 - u)(6 - v)(b -w) ^ , . . . 

a — b 

2 _ ( M — ^)(^ — w) 2 _ (P — U)(V — to) 

** ~ 4(o - u)(6 - u)' Qv ~ 4(a - v)(fi - v)' 

2 — ( W ~ U )( W ~ V ) 

qw - 4(a - w)(6 - tu)' 
A Coordinate System Involving Elliptic Functions 

x = u dn(t>, k) sn(w, fc'), 2/ = u sn(t;, &) dn(w, A;'), 

z = u cn(t>, Aj) cn(w, &'), k 2 + A;' 2 = 1. 
ql = 1, ffj = tfi = ^ 2 {fc 2 cn 2 (;;, t) + A;' 2 cn 2 (w, *')}. 

For a discussion of the elliptic functions dn, sn, and en see 
W. F. Osgood, "Advanced Calculus," Chap. IX, or E. P. Adams, 
"Smithsonian Mathematical Formulae-" p. 245. 



APPENDIX V 

THE EVALUATION OF THE MUTUAL ELECTROSTATIC 

ENERGY OF TWO SPHERICALLY SYMMETRICAL 

DISTRIBUTIONS OF ELECTRICITY WITH 

EXPONENTIAL DENSITY FUNCTIONS 

In Section 23b there occurs the integral 



Ze 2 f Ce^ur^n. . 

- I I —7- ClTidT 2, 



327r 2 aoJ J P12 

in which pi = 2Zri/a and dri = p 2 dpi sin &id&id<pi, with similar 
expressions for p 2 and dr 2 , n, #1, <Pi and r 2 , #2, <p 2 being polar 
coordinates for the same system of axes. The quantity P12 
represents 2Zri 2 /a , in which r i2 is the distance between the 
points r x , #1, <pi and r 2 , # 2 , <p 2 . 

This integral (aside from the factor Ze 2 /327r 2 a ) represents the 
mutual electrostatic energy of two spherically symmetrical 
distributions of electricity, with density functions e~ p i and 
e~ p *, respectively. It can be evaluated by calculating the 
potential due to the first distribution, by integrating over 
dri, and then evaluating the energy of the second distribution 
in the field of the first. 

The potential of a spherical shell of radius pi and total charge 
btrp\er p xdpi is, 1 at a point r, 



and 



4irple- p idpi • — for r < pi 
Pi 



4rp 2 i e~ p idp 1 • - f or r > pi; 
r 



that is, the potential is constant within the shell and has the 
same value outside of the shell as if the entire charge were 
located at the origin. 

1 See, for example, Jeans, "Electricity and Magnetism," Cambridge 
University Press, Cambridge, 1925, Sec. 74. 

446 



APPENDIX V 447 

The potential of the complete distribution is hence 

<t>(r) = — I e- p ip\dpi + 4tt I e~ p ipidpi, 
which is found on evaluation to be 

*(r) =^{2-e-(r + 2)}. 
The integral / then has the value 

1 = 32^J *<")^«*t, 

Ze 2 f " 
= 2^1 {2 - e-'^pj + 2)je-"»p 2 rf P8 , 

which gives on integration 



APPENDIX VI 

NORMALIZATION OF THE ASSOCIATED LEGENDRE 
FUNCTIONS 

We can obtain the orthogonality property of the functions 
P\ m] (z) and P\T^(z) as follows: Multiply the differential equation 
19-9 satisfied by P\ m] (z) by P[? ] (z) and subtract from^his the 
differential equation satisfied by P\? ] (z) multiplied by P[ m{ (z). 
The result is the relation 

= \v<y + l) - l(i + i)}P\rP[ m K 

If we integrate this between the limits —1 and 1, we obtain the 
result 

{l'(V + 1) - 1(1 + \))f^P\7*{z)P\*(z)dz 

Therefore, if V ^ I, 

j^Pf(z)P\^(z)dz = 0. (1) 

This result is true for any value of m, so it is also true for the 
Legendre functions Pi(z), since Pi(z) = P?(z). 

We can now obtain the normalization integral for the Legendre 
polynomials. Replacing I by I — 1 in Equation 19-2 gives the 
equation 

Pi(z) = j{(2l - i)riViOO - (J - DPi-tOO). 

Using this and the orthogonality property just proved, we obtain 
the relation 

448 



APPENDIX VI 449 

J + 1 27-1 f + 1 

i {Pi{z))Hz = fi T - i J_ i Pi-i(z)zPi(z)dz. 

Equation 19-2 can be written in the form 

zPiiz) = 2TT1 {(Z + 1)Pz + l(z) + lPl ~^> 
so that, again employing the orthogonality property, we get 

This process can be repeated until the relation 



x 



{iM*)P<k = 



(21 - l)(2l - 3)(2l - 5) • • -31 f +1 

(21 + l)(2l - 1)(21 - 3) ■ • • 5 -3j_i ,r,w ' 

= 2nnXr {Po(i5)12<fo 

is obtained. PoOO is by definition (Eq. 19-1) the coefficient of 
t° in the expansion of (1 — 2tz + t 2 )~~M in powers of t. It is 
therefore equal to unity, so that 



£>K*)^ = ^£> = 



21 + 1 



(2) 



To obtain the normalization integral for the associated 
Legendre functions we proceed as follows. 1 By differentiating 
Equation 19-7 and multiplying by (1 — z 2 )^ we obtain 

(1 - ^Mfi^al = (1 - ^ 2 I^P^)" 
M-l J|«| 

M*d - * 2 ) 2 ^ra p '(«) = p l ml+1 (2) - M*(i - « 2 )-^l ml («). 



I 2 

+ 



Transposing, squaring, and integrating gives 

xr {pim,+i(2)} ^=xr[ (i -^{^} 2 

1 Whittaker and Watson, "Modern Analysis," Sec. 15-51. 



450 APPENDIX VI 

f + 1 mW 

+ J-i 1 -2 



where integration by parts 1 has been employed to obtain the 
first two terms of the last line. 

If we now use the differential equation 19-9 for P\ ml (z) to 
reduce the first term of the last line, we obtain, after combining 
terms, the result 

f^iPP+KWdz = (I - M)(Z + \m\ + l)f^{P\-Kz)} 2 dz. 
We can continue this process and thus obtain 

f^{P\ m Kz)} 2 dz = (1 - \m\ + l)(l - \m\ + 2) • • • I 

(I + \m\)(l + |m| - 1) • • • (Z + l)f +*&&)}*&, 
so that 



x 



-i {n {)] 21 + 1(1- \m\)f 

where we have used the result of Equation 2. 

/r dP i r* 

udv — uv — I vdu, we set u = (1 — z 2 )— — , 

dP\ mS 

dv — in order to reduce the first term, and u = z, 

dz 

jp|m| 

dv - 2Pl m| — l —dz « d(P' z m, p 
dz 

to reduce the second term. The term in uv vanishes, In the first case because 
(1 — z 2 ) is zero at the limits, and in the second case because P[ m| (2) is zero 
at the limits, if m ^ 0. 



APPENDIX VII 

NORMALIZATION OF THE ASSOCIATED LAGUERRE 
FUNCTIONS 

In order to obtain Equation 20-10, we make use of the generat- 
ing function given in Equation 20-8, namely 



^^(pLr.r-iv. 



1-u 



u.( P , u) - 2i~W ur s ^'( l-u)-" "'- 

r = « 

Similarly let 



p» 



y.y, .) - 2^ " - (-v ( T^.-- 



* = « 



Multiplying these together, introducing the factor e~ p p 8+1 f and 
integrating, we obtain the equation 

00 

J[ Vv+><7.(p, u)F £ ( p , t,)dp = ^^§~ e-' P >"L' r {p)L- t { P )dp 

= (1 - w )«+» = ( * + 1)!(1 ~ w ~ " + ™>) 

^(g + fc+1)! 

where we have expanded (1 — uv)~"~ 2 by the binomial theorem. 1 
The integral we are seeking is (r!) 2 times the coefficient of 
(uv) r in the expansion, which is 

/• 00 

x For the value of the integral I p*+ l e~ aP dp see Peirce's " Table of 

Integrals." 

451 



452 APPENDIX VII 

We, + i)ii < r + 1 > ! + zl 1 

V.) ^ -t- i;.^ (r _ s)|(s + 1}1 -f- (r _ s _ 1)f(s + 1)f | 

(r!) 3 (2r - g + 1) 
(r - s) ! 

In order to obtain the integral of Equation 20-10 we must put 
r = n + J and s = 21 + 1, yielding the final result 



x 



'e-v<+w + v(p)p<* P = ( yi w +.ffi 



APPENDIX VIII 
THE GREEK ALPHABET 



A, a . 


. . Alpha 


N, v . 


. . Nu 


B,0 . 


. . Beta 


s, f . 


. Xi 


r, 7 • 


. . Gamma 


0,0 . 


. Omicron 


A, 5 . 


. . Delta 


n, t . 


. Pi 


E,« . 


. . Epsilon 


p,p . 


. BAo 


z,r • 


. . Zeta 


2,<r . 


. Sigma 


H, v ■ 


. . Eta 


T,r 


. Taw 


0, #, 


. . Theta 


T, v . 


. Upsilon 


I,i . 


. . Iota 


% <p 9 4 • 


. Phi 


K, K . 


. . Kappa 


X, x • • 


. CW 


A, X . 


. . Lambda 


*,* • 


. Psi 


M,/. . 


. . Tkfw 


12, co . 


. Omega 



453 



INDEX 



Absorption of radiation, 21, 299 
Accessible wave functions, 396, 397 
Action, 25 
Action integrals, 29 
Activation energy, 412 
Adams, E. P., 201 
Adsorption, unactivated, 388 
Alkal^atom spectra, £07 
Alternating intensities in band spec- 
tra, 356 
Amplitude equation, 56 

in three dimensions, 86 
Amplitude functions, definitions of 

58 
Amplitudes of motion, 286 
Anderson, C, 209 
Angular momentum, of atoms, 237 
conservation of, 11 
of diatomic molecules. 265 
of elefnon spin, 208 
of h^pgen atom, 147 
properties of, 425 
of symmetrical top molecule, 280 
Antisymmetric wave function, defi- 
nition of, 214 
Approximate solution of wave equa- 
tion, methods of, 191 
(See also Wave functions.) 
Approximation by difference equa- 
tions? 202 
Aromatic carbon compounds, ener- 
gies of, 379 
Associated Laguerre functions, 
normalization, 451 
polynomials, 131 
table of, 135 
Associated Legendre functions, 127 
table of, 134 



Asymptotic solution of wave equa- 
tion, 68 
Atanasoff, J. V., 228 
Atomic energy states, semi-empirical 

treatment, 244 
Atomic terms, Hund's rules for, 246 
Atomic wave functions, 250 
Atoms, with many electrons, 230$". 

variation treatments for, 246 
Average values, in quantum mechan- 
ics, 89 
of dynamical quantities, 65 
of r* for hydrogen atom, 144 
Azimuthal quantum number, 120 



B 



Bacher, R. F., 258 
Balmer formula, 27 
Balmer series, 43 
Bartholomew E., 310 
Bartlett, J. H., Jr., 254 
Beach, J. Y., 385 
Bear, R. S., 415 
Beardsley, N. F., 249 
Benzene, plane structure of, 411 

structure of, 378 
Beryllium atom, wave functions for, 

249 
Bichowsky, R., 208 
Birge, R. T., 41, 336, 439 
Black, M. M., 254 
Black body, 25 
B6cher, M., 419 
Bohr, N., 26, 36, 112 
Bohr frequency rule, 27 
Bohr magneton, 47 
Bohr postulates, 26 
Boltzmann distribution law, 399 
Boltzmann statistics, 219 



455 



456 



INDEX 



Bonds, chemical, types of, 362 
Bond wave functions, 374 
Bonhoeffer, K. F., 358 
Born, M., 49, 51, 112, 260, 364, 416, 

417, 425, 436 
Born-Oppenheimer principle, 260 
Bose, S. N., 403 

Bose-Einstein distribution law, 404 
Bose- Einstein statistics, 219, 402 
Bowen, I. S., 208 
Brackett series, 43 
Bragg equation, 35 
Brester, C. J., 290 
Brillouin, L., 198, 412 
Brockway, L. O., 379 
Brown, F. W., 254 
Burrau, , 333, 340 
Byerly, W. E., 24 



Canonical form of equations of 
motion, 16 

Canonical set of structures, 375 

Carbon atom, tetrahedrai, 364, 377 
variation function for, 249 

Carbon compounds, aromatic, ener- 
gies of, 379 

Catalysis of ortho-para conversion, 
358 

Central-field approximation for 
atoms, 230, 250 

Characteristic energy values, 58 

Characteristic functions, definition 
of, 58 

Characteristic value equation, defi- 
nition of, 58 

Characteristic values, approximate, 
180 

Chemical bonds, types of, 362 

Classical expressions, significance of, 
55 

Classical mechanics, 2ff. 

as an approximation to quantum 

mechanics; 198 
and high quantum states, 76 

Classical statistics, 219 



Coefficient, of absorption, definition 

of, 300 
of induced emission, definition of, 

300 
of spontaneous emission, defini- 
tion of, 300 
Coefficients in secular equation ,for 

molecule, 376 
Commutation rules, 420 
Completed shells of electrons, 234 
Completeness of sets of orthogonal 

functions, 154 
Complex conjugate wave function, 

63, 88 
Complex molecules, 366^. 
Component of angular momentum, 

definition of, 12 
Compton, A. H., 35 f 

Conditionally periodic systems, defi- 
nition of, 29 
Conditions on wave functions, 58 
Condon, E. U., 54, 82, 108, 246, 258, 

310, 312, 432 
Configuration, electronic, definition 

of, 213 
Configuration space, definition of, 

59 
Conjugate wave function, complex, 

63 
Conservation, of angulaaenomen- 

tum, 11 
of energy in quantum mechanics, 

75 
Conservative system, definition of , 16 
Constant of the motion, definition 

of, 12, 422 
Continuous sets of energy levels, 58 
Coolidge, A. S., 188, 249, 349, 353, 

364, 374, 415 
Coordinates, curvilinear, 103 
cyclic, 108 
generalized, 6 
ignorable, 108 
for molecules, 389 
normal, definition of, 287 
Correspondence principle, 29 
Coulomb integral, 212, 371 
Coupled harmonic oscillators, 397 



INDEX 



457 



Courant, R., 91, 120, 157, 192, 202 
Cross, P. C, 282 
Crystal, diffraction by, 34 
Crystals, rotation of molecules in, 

290 
Curvilinear coordinates, 103 
Cyclic coordinates, 108 



D 



Darrow, K. K, 54, 83, 403, 412 
Darwin, C. G., 209, 210, 432 
De Broglie, L., 49, 93 
De Broglie wave length, 35 
Debye, P., 26, 383, 408, 412 
Degeneracy, exchange, 230 
>^jtal, 367 
ipatial, 233 

flip 367 
Degenerate energy levels, 73, 166 

Degenerate states, 31, 100 

Degeneration of gases, 405 

Del squared, operator, 85 

Dennison, D. M., 275, 279, 293, 357 

Determinants, certain properties of, 

174 
Determjuiant-type wave functions, 

21^232 

DiagdHporm for secular equation, 

Diagonal matrices, 421 
Diagonal-sum theorem, 239 
Diatomic molecule in old quantum 
theory, 32 
rotation and vibration of, 263 
selection rules and intensities for, 
309 
Dickinson, B. N., 331 
Dieke, 6. I|., 282 
Dielectric constant, 408 
of diatomic dipole gas, 408 
and polarizability, 227 
Difference equations approximating 

wave equation, 202 
Differential equation for Legendre 
polynomials, 127 
standard form for, 109 



Diffraction by a crystal in old 

quantum theory, 34 
Dipole interaction, 384 
Dipole moment, electric, definition 

of, 303 
Dipole radiation, definition of, 23 
Dirac, P. A. M., 49, 112, 209, 210, 

256, 294, 299, 403, 417, 433, 436 
Dirac equations and electron spin, 

209 
Directed valence, 377 
Discrete sets of energy levels, 58 
Dissociation energy, of hydrogen 

molecule, 349, 352 
of hydrogen molecule-ion, 336 
Distribution law, Boltzmann, 399 
Doi, S., 179 

Doublets in alkali atom spectra, 207 
Duane, W., 35 
Dunham, J. L., 198 
Dunkel, M., 346 
Dunn, C. G., 254 



E 



Eckart, C, 49, 180, 222, 247, 275, 

417 
Edlen, B., 225 
Ehrenfest, P., 36 
Eigenfunction, definition of, 58 
Eigenwert, definition of, 58 
Einstein, A., 25, 300, 403 
Eisenhart, L. P., 105 
Eisenschitz, R., 386 
Electric dipole moment, definition 

of, 303 
Electric moment, of molecules, 411 

of a system, definition of, 23 
Electron, spinning, 207 
Electron affinity of hydrogen, 225 
Electron densities for atoms> 257 
Electron density for lithium, 249 
Electron diffraction by a crystal, 34 
Electron distribution for hydrogen 

molecule-ion, 337 
Electron distribution function for 

lithium, 249 
Electron-pair bond, 362 



458 



INDEX 



Electron-pairing approximation, 374 
Electron-spin functions for helium, 

214 
Electron-spin quantum number, 208 
Electronic configuration, definition 

of, 213 
Electronic energy function for dia- 
tomic molecules, 266 
Electronic energy of molecules, 259 
Electronic states, even and odd, 313 
Electronic wave function for mole- 
cule, 261 
Elliptic orbit, equation of, 38 
El-Sherbini, M. A., 179 
Emde, 343 

Emission of radiation, 21, 299 
Empirical energy integrals for 

atoms, 244 
Energy, of activation, 412 
of classical harmonic oscillator, 5 
correction to, first-order, 159 

second-order, 176 
and the Hamiltonian function, 16 
of hydrogen molecule-ion, 336 
kinetic, definition of, 2 
of molecules, separation of, 259 
potential, definition of, 2 
of resonance in molecules, 378 
of two-electron ions, 225 
values of, for atoms, 246 
Energy level, lower limit for, 189 

lowest, upper limit to 181 
Energy levels, 58 
approximate, 180 
for diatomic molecule, 271, 274 
for harmonic oscillator, 72 
for plane rotator, 177 
for symmetrical top molecule, 280 
vibrational, of polyatomic mole- 
cule, 288 
Epstein, P. S., 36, 179, 191 
Equation, homogeneous, 60 
Equations of motion, in Hamil- 
tonian form, 14 
in Lagrangian form, 8 
Newton's, 2 
Ericson, A., 225 



Ethane molecule, free rotation in, 
280 

Eucken, A., 26 

Eulerian angles, 276 

Even and odd electronic states, 313 

Even and odd states of molecules, 
354 

Even and odd wave functions for 
molecules, 390 

Exchange degeneracy, 230 
integral, 212, 372 

Excited states, of helium atom, 225 
of hydrogen molecule, 353 
of hydrogen molecule-ion, 340 
and the variation method, 186 

Exclusion principle, 214 

Expansion, of l/r»y, 241 
in powers of h, 199 
in series of orthogonal fun<Maoi|6, 
151 

Eyring, H., 374, 376, 414 



Factorization of secular equation 
for an at uii, 235 

Farkas, A., 358, 414 

Fermi, E., 257, 403 

Fermi-Dirac distribution law. -*ii3 

Fermi-Dirac statistics, 219 

Field, self -consistent, 250# 

Fine structure, of hydrogen spec- 
trum, 207 
of rotational bands, alternating 
intensities in, 356 

Finkelstein, B. N., 331 

Fock, V., 252, 255 

Force, generalized, 7 

Force constant, definition of, 4 

Forces between molecules, 3$3 

Formaldehyde, rotational fine struc- 
ture for, 282 

Formulas, chemical, meaning of, 380 

Fourier series, 153 

Fowler, R. H., 412 

Franck, J., 310 

Franck-Condon principle, 309 

Frank, N. H., 275 



INDEX 



459 



Free particle, 90 

Free rotation in molecules, 280, 290 

Frenkel, J., 83, 437 

Frequency, of harmonic oscillator, 5 

of resonance, 320 
Friedrichs, 202 
Fues, E., 274 

Fundamental frequency, definition 
of, 290 



G 



g factor for electron spin, 208 

Geib, K. H., 414 

General solution of wave equation, 

57 
Gerifiral theory of quantum mechan- 
] ics, 416jf. 

Generalized coordinates, 6 
GeneHdized forces, 7 
Generalized momenta, definition of, 

14 
Generalized perturbation theory, 

191 
Generalized velocities, 7 
Generating function,' IJfPr'associated 

Laauerre polynomials, 131 
for aitociate *, Legendre functions, 

deMkn of, 77 

for^Haerre polynomials, 129 

for I^ejjendre polynomials, 126 
Gentile, G., 361 
Ginsburg, N., 246 
Gordon, W., 209 
Goudsmit, S., 207, 208, 213, 221, 

227, 237, 246, 257, 258, 313 
Gropper, L., 405 
GroupJ completed, of electrons, 125 

deration of, 231 
Group theory and molecular vibra- 
tions, 290 
Guillemin, V., 247, 332, 353 



Half-quantum numbers, 199 
Hamiltonian equations, 16 



Hamiltonian form of equations of 

motion, 14 
Hamiltonian function, definition of, 
16 
and the energy, 16 
and the wave equation, 54 
Hamiltonian operator, 54 
Harmonic oscillator, average value 
of x\ 161 
classical, 4 

in cylindrical coordinates, 105 
energy levels for, 72 
in old quantum theory, 30 
perturbed, 160 
selection rules and intensities for, 

306 
three-dimensional, in Cartesian 

coordinates, 100 
wave functions, mathematical 

properties of, 77 
in wave mechanics, 67jf. 
Harmonic oscillators, coupled, 315Jf., 

397 
Harmonics, surface, 126 
Harteck, P., 358, 414 
Hartree, D. R., 201, 224, 250, 254, 

255 
Hartree, W., 255 
Hass6, H. R., 185, 228, 385, 387 
Heat, of activation, 412 

of dissociation, of .hydrogen mole- 
cule, 349, 352 
of hydrogen molecule-ion, 336 
Heat capacity, of gases, 408 

of solids, 26 
Heats of sublimation, 388 
Heisenberg, W., 48, 112, 209, 210, 
226, 318, 416, 417, 428, 432, 437 
Heisenberg uncertainty principle, 428 
Heitler, W., 340, 361, 364 
Helium, solid, equilibrium distance 

in, 362 
Helium atom, 210 
accurate treatments of, 22 
excited states of, 225 
ionization potential of, 221 
normal state of, by perturbation 
theory, 162 



460 



INDEX 



Helium atom, polarizability of, 226 

resonance in, 321, 324 

with screening constant func- 
tion, 184 

spin functions of, 214 

stationary states of, 220 
Helium molecule-ion 358, 367 
Hermite orthogonal functions, 80 
Hermite polynomials, 77, 81 

recursion formula for, 71 
Hilbert, D., 91, 120, 157, 192 
Hill, E. L., 83 
Hiyama, S., 179 
Homogeneous equation, definition 

of, 60 
Homogeneous linear equation, solu- 
tion of, 169 
Hooke's forces in molecules, 282 
Hooke's law constant, 4 
Hooke's potential energy for dia- 
tomic molecules, 267 
Horowitz, G. E., 331 
Houston, W. V., 221 
Huckel, E., 346, 365, 379, 381 
Hultgren, R., 377 
Hund, F., 346, 381, 390 
Hund's rules for atomic terms, 246 
Hydrogen atom, 112 

continuous spectrum of, 125 

electron affinity of, 225 

energy levels of, 42, 124 

momentum wave functions of, 
436 

normal state of, 139 

in old quantum theory, 36Jf. 

old-quantum-theory orbits, 43 

perturbed, 172 

polarizability of, 185, 198, 205 

selection rules for, 312 

solution of r equation, 121 

solution of theta equation, 118 

solution of wave equation, 113 

spectrum of, 42 

Stark effect of, 178, 195 
Hydrogen atoms, three, 368, 414 
limiting cases for, 372 
wave functions for, 368 

Van der Waals forces for, 384 



Hydrogen chloride, absorption band 

of, 33 
Hydrogenlike radial wave functions, 

discussion of, 142 
Hydrogenlike wave functions, 132 

discussion of angular part of, 146 
Hydrogen molecule, 340Jf. 

excited states of, 353 
Hydrogen molecule-ion, 327j(f. 
Hydrogen spectrum, fine structure 

of, 207 
Hydrogen sulfide, rotational fine 

structure for, 282 
Hylleraas, E. A., 222, 225, 226, 335, 

337, 340, 353, 436 
Hypergeometric equation, 278 



Identity operation, definition of, 231 
Ignorable coordinates, 108 
Independent sets of wave functions, 

216 
Index of refraction and polarizabil- 
ity, 227 
Indicial equd ion, 109 
Induced emission, 300 i 

Infinite determinant*, soljpon of, 

339 
Infinity catastrophe, 60 
Inglis, D. R., 246 
Ingman, A. L., 224 
Integrals, energy, for atoms, 239 
involved in molecular-energy cal- 
culations, 370 
involving determinant-type wave 
functions, 239 
Intensities, for diatomic molecule, 
309 
for harmonic oscillator, 30p 
for surface-harmonic wave func- 
tions, 306 
Interaction, of helium atoms, 361 x 
of hydrogen atom and molecule, 
373 
Interatomic distance in hydrogen 
molecule, 349 



INDEX 



461 



Interchange integrals, definition of, 
212 

Invariance of equations of motion, 7 

Inverse permutation, definition of, 
231 

Ionic 'contribution to bonds, 364 

Ionic structures for hydrogen mole- 
cule, 345 

Ionization energy of two-electron 
s "ions, 225 

Ionization potential for helium, 221 

Ionization' potential for lithium, 247 

Ishida, Y., 179 

Islands, 376 

Ittmann. G. P., 282 



JaiHi G., 335, 340 

Jahjke, E., 343 

James, H. M., 188, 249, 333, 349, 

353, 362, 374, 415, 
Jeans, J. H., 24, 241 
Johnson, M„, / T 5 
Johnson, M. h!, Jr.; 
Jordlfe P., 49, 112, &09, 416, 417, 
433, 436 



2,Soa 4 



K 



£arman, T. von, 26 
om, W. H., 383 
ku!6 structures, resonance of, 378 
ner, G. W., 222 
Kemble, E. C., 83, 353 
Kennard, E. H., 432 
Kerr's area law, 37 
KifflLll, G. E., 203, 376, 415 
Kine% .^^rgy, definition of, 2 

Kirkwood/lKjL, 191, 228, 385, 387 
XistiakowsB^ltB . , 282 
Klein, 0., 282 W 
Kohlrausch, K^jflT. F., 293 
Kramers, H. A., 198, 282 
Kronig, R. de L., 276, 293, 390 



La Coste, L. J. B., 280 

Lagrange's equations of motion, 8 

Lagrangian function, definition of, 3 

Laguerre polynomials, 129 

A-type doubling, 390 

LandS, A., 208 

g factor for electron spin, 208 

Langevin, P., 412 

Laplace operator, in Cartesian coor- 
dinates, 85 
in curvilinear coordinates, 104 

Legendre functions, 125 

Lennard-Jones, J. E., 191, 206, 290, 
340, 381, 388 

Lewis, G. N., 340, 377 

Lewy, H., 202 

Light (see Radiation) 

Linear combinations and resonance, 
320 

Linear independence of wave func- 
tions, definition of, 166 

Linear momentum, average, of elec- 
tron in hydrogen atom, 146 

Linear variation functions, 186 

Lithium atom, electron distribution 
function for, 249 
wave functions for, 247 

London, F., 340, 361, 364, 383, 386, 
387, 388, 413 

Loney, S. L., 24 

Lyman, T., 222 

Lyman series, 43 

M 

MacDonald, J. K. L., 188, 189, 353 
McDougall, J., 254 
MacMillan, W. D., 24 
Magnetic moment, 412 

of electron spin, 208 

of hydrogen atom, 147 

orbital, 47 
Magnetic quantum number, 40, 117 
Magnetic susceptibility, 412 
Magneton, Bohr, 47 
Majorana, E., 353, 359 



462 



INDEX 



Many-electron atoms, 230/. 

Margenau, H., 386, 387 

Matossi, F., 293 

Matrices, 417jf. 

Matrix algebra, 417 

Matrix mechanics, 416jf. 

Maximal measurement, 422 

Mayer, J. E., 229 

Mayer, M. G., 229 

Measurements, prediction of results 

of, 66 
Mecke, R., 282 
Millikan, R. A., 208 
Modes of vibration of molecules, 287 
Mole refraction, definition of, 227 
Molecular energy levels, 259 
Molecular orbitals, 346 

method of, 381 
Molecular wave functions, sym- 
metry properties of, 388 
Molecule, diatomic, selection rules 

and intensities of, 309 
Molecules, complex, 366jf. 
diatomic, rotation and vibration 

of, 263 
polyatomic, rotation of, 275 

vibration of, 282 
quantum number A in, 390 
Moment of inertia, 269, 275 
Momenta, generalized, definition of, 

14 
Momentum, angular, conservation 
of, 11 
average linear, of electron in 

hydrogen atom, 146 
operator, 54 
Momentum wave functions, 436 
Morse, P. M., 54, 82, 108, 249, 272, 
312, 340, 437 
function for diatomic molecules, 
271 
Mott, N. F., 83 
Mulliken, R. S., 346, 381 
Multiplication of permutations, defi- 
nition of, 231 
Multiplicity of atomic terms, 220 



N 

Negative states, 392 

Nernst, W., 26 

Neumann, J. v., 437 

Newton's equations, 2 

Nielsen, H. H., 280, 282 

Niessen, K. F., 327 

Non-degenerate energy levels/defini- 
tion of, 73 

Normal coordinates, 282 
mode of vibration ©^definition 
of, 287 

Normalization, of amplitude func- 
tions, 89 
of wave functions, 64 
for a continuum, 92 

Nuclei r s7pin for hydrogen, out 

Nu t'ca# mmetry of electronic irave 
i vjtions for molecules, 39t r 

Nuclear wave function for molecule, 
263 

Numerical integration, 201 



Old quantum diieory, as an approxi- 
mation to I quantum mectyjuiics, 
198 
decline of, 48 

One-electron bond, 362 1 

Operator, for Hamiltonian>Y 
for momentum, 54 

Operators for dynamical quantiti< 
66 

Oppenheimer, J. R., 260 

Orbit, classical, of three-dimen- 
sional oscillator, 11 

Orbital, definition of, 137 

Orbital degeneracy, 367 

Orbitals, molecular, 381 

Orbits, significance of, intouantum 
mechanics, 141 

Ortho helium, 221 

Ortho hydrogen, 35JJW6& 

Orthogonal curvilralar coordinate 
systems, 441 >, 

Orthogonal functions, a convenient 
set of, 195 



IhDEX 



463 



Orthogonal functions, expansions in 
terms of, 151 

Orthogonal transformation, defini- 
tion of, 288 

prthogonality of wave functions, 64, 
89, 441 
illation of molecules in crystals, 

Dr, classical, in polar coor- 
es, 9 

tl harmonic, 4 
LarmcSnjfc, K in cylindrical coor- 
dinates, 105 
in old quantum theory, 30 
perturbed, 160 

tjiree-dimensional, in Cartesian 
,' coordinates, 100 
in wave mechanics, 
GUfe-dimensional clap 



Para hydro' sn, 357, 408 
Parhelium, $M 
Particl^pfl^irjfeOo^^ 
i in field i vee sp^^jlHL. 
t&vniri quantum thlpry, 33 

a, F., 222 
Kfr»**fcSb series, 43 > 




L 58, 112, 209, 210, 219, 
f327, 403 

iion principle for protons, 357 
Mlling, L., 227, 256, 257, 290, 327, 
347, 359, 362, 364, 365, 376, 379, 
385, 436 
Bakeris, C. L., 272 
P<4zer, H., 414 
Pe&etration, of the core, 213 

r non-classical region, 75 
Pe|^, W. G., 379 
Pe^&Syiaui operator, 231 
Perikutatffl||» even and odd, defini- 
tion ( 
Perturbatfd&Aiause of transitions, 
294jf. 
definition of. 156 
theory of. 15 



Perturbation, theory of, first-order 
for a degenerate level, 165 
for non-degenerate levels, 156 
generalized, 191 
involving the time, 294^. 
second-order, 176 
approximate, 204 
Phase integrals in quantum mechan- 
ics, 200 
Phases of motion, 286 
Photochemistry, 26 
Photoelectric effect, 25 
Photon, 25 

Physical constants, values of, 439 
Physical interpretation, of harmonic 
oscillator functions, 73 
of wave equation, 298 
of wave functions, 63, 88 
Pike, H. H. M., 290 
Placzek, G., 290, 293 
Planck, M., 25 
Planck's constant, 25 
Planck's radiation law, 301 
Plane rotator. Stark effect of, 177 
Podolanski, J., 335 
Podolsky, B., 276, 436 
Polanyi, M., 374, 388, 414 
Polar coordinates, spherical, 9 
Polarizability, and dielectric con- 
stant, 227 
of helium atom, 226 
of hydrogen atom, 185, 198, 205 
and index of refraction, 227 
of plane rotator, 178 
and van der Waals forces, 387 
Polarization, of emitted light, 308 

of a gas, 227 
Polarization energy, of hydrogen 
molecule, 349 
of hydrogen molecule-ion, 332 
Polynomial method of solving wave 

equation, 68 
Positive states, 392 
Postulates of wave mechanics, addi- 
tional, 298 
PoBtulatory basis of physics, 52 
Potential energy, average, for hydro- 
gen atom, 146 



464 



INDEX 



Potential energy, definition of, 2 

Potential function for diatomic 
molecules, 267 

Power-series method of solving wave 
equation, 69 

Present, R. D., 353 

Principal axes of inertia, definition 
of, 275 

Probability, of distribution func- 
tions, 63 
of stationary states, 298 
of transition, 299 

Proper functions, definition of, 58 

Properties of wave functions, 58 



Quadratic form, minimization of, 203 
Quadrupole interaction, 384 
Quadrupole moment, definition of, 

23 
Quantization, rules of, 28 

spatial, 45 
Quantum of energy, 25 
Quantum number, azimuthal, 40, 
120 

electron-spin, 208 

A in molecules, 390 

magnetic, 40, 117 

orbital angular momentum, 237 

radial, 124 

rotational, 33 

spin for atoms, 237 

total, 41, 124 
Quantum numbers, 87, 124 

half-integral, 48 

in wave mechanics, 62 
Quantum statistical mechanics, 219, 

395jf. 
Quantum theory, history of, 25 

old, 25jf. 
Quantum weight, definition of, 100 

R 

Rabi, I. I., 276 
Rademacher, H.. 276 



Radial distribution function for 

hydrogen atom, 140 
Radial quantum number, 124 
Radiation, emission and absorption 
of, 21, 299 

of kinetic energy, 314 

Planck's law of, 301 
Rate of chemical reactions, 412 
Reaction rates, 412 
Recursion formula, definition <ff, 70 

for Hermite polynomials, 71 

for Legendre polynomials, 126 
Reduced mass, 18, 37 
Regular point, definition of, 109 
Reiche, F., 276 
Relativist change of mass of 

elep^ T 209 
Relati stic doublets, 209 
Representation, 423 
Repulsion of helium atoms, 361 
Repulsive states of hydrogen mol- 
ecule, 354 
Resonance, among bond structures, 
377 

class\ d, 31*) 

dei ^;. a , neo— - 

energy, <>*. „ r 

frequency oi, 320 v ; 

in the hydrog ^n molecule-w^ 830 
332 ^ 

integrals, definition of, ! 

phenomenon, 214 - 

quantum-mechanical, 314, 318 
Restrictions on wave functions, 58 
Richardson, O. W., 336 
Richardson, R. G. D., 202 
Rigid rotator, wave functions for, 

271 
Ritz combination principle, 27 
Ritz method of solution, 189 
Ritz variation method, 189 
Robertson, H. P., 105, 432 
Robinson, G., 201 
Rosen, N., 332, 349 ^ 

Rotation, of diatomjjf molecules, 2153 

of molecules in crystals, 290 

of polyatomic ijaolecules, 275 
Rotational energy of molecules, 259